Applying the Cubic Formula

Oct 13, 2025 by sorusgentrification.com 110 views

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Introduction

Solving cubic equations can be a challenging task, especially when dealing with complex numbers. In this article, we will explore the process of finding real and complex solutions to the cubic equation $x^3 - 3x^2 - 10x + 24 = 0$ . We will use various techniques, including factoring, the Rational Root Theorem, and the use of the cubic formula.

The Cubic Formula

The cubic formula is a general method for solving cubic equations of the form $ax^3 + bx^2 + cx + d = 0$ . The formula is given by:

x = \frac{-b \pm \sqrt[3]{-q + \sqrt{q^2 + p^3}}}{3a}

where $p = \frac{3ac - b^2}{3a^2}$ and $q = \frac{2b^3 - 9abc + 27a^2d}{27a^3}$ .

Applying the Cubic Formula

To apply the cubic formula, we need to calculate the values of $p$ and $q$ . In this case, we have:

a = 1, b = -3, c = -10, d = 24

Substituting these values into the formulas for $p$ and $q$ , we get:

p = \frac{3(1)(-10) - (-3)^2}{3(1)^2} = \frac{-30 - 9}{3} = -13

q = \frac{2(-3)^3 - 9(1)(-10)(24) + 27(1)^2(24)}{27(1)^3} = \frac{-54 + 4320 + 648}{27} = \frac{5914}{27}

Simplifying the Cubic Formula

Now that we have the values of $p$ and $q$ , we can simplify the cubic formula:

x = \frac{-(-3) \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\left(\frac{5914}{27}\right)^2 + (-13)^3}}}{3(1)}

x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\frac{34941196}{729} + 2197}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\frac{34941196 + 1598593}{729}}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\frac{36539789}{729}}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \frac{36539789}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{\frac{-5914 \cdot 27 + 36539789}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{\frac{-159414 - 36539789}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{\frac{-36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3}

x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3<br/> # **Frequently Asked Questions about Finding Real and Complex Solutions to a Cubic Equation** ===========================================================

Q: What is the cubic formula?

A: The cubic formula is a general method for solving cubic equations of the form $ax^3 + bx^2 + cx + d = 0$ . The formula is given by:

x = \frac{-b \pm \sqrt[3]{-q + \sqrt{q^2 + p^3}}}{3a} </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>a</mi><mi>c</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">p = \frac{3ac - b^2}{3a^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">c</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><mn>9</mn><mi>a</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>27</mn><msup><mi>a</mi><mn>2</mn></msup><mi>d</mi></mrow><mrow><mn>27</mn><msup><mi>a</mi><mn>3</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">q = \frac{2b^3 - 9abc + 27a^2d}{27a^3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span><span class="mbin mtight">+</span><span class="mord mtight">27</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight">d</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p> <h2><strong>Q: How do I apply the cubic formula?</strong></h2> <hr> <p>A: To apply the cubic formula, you need to calculate the values of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>. This involves substituting the coefficients of the cubic equation into the formulas for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>.</p> <h2><strong>Q: What are the steps to simplify the cubic formula?</strong></h2> <hr> <p>A: To simplify the cubic formula, you need to follow these steps:</p> <ol> <li>Calculate the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> using the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>a</mi><mi>c</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">p = \frac{3ac - b^2}{3a^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">c</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>Calculate the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span> using the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><mn>9</mn><mi>a</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>27</mn><msup><mi>a</mi><mn>2</mn></msup><mi>d</mi></mrow><mrow><mn>27</mn><msup><mi>a</mi><mn>3</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">q = \frac{2b^3 - 9abc + 27a^2d}{27a^3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span><span class="mbin mtight">+</span><span class="mord mtight">27</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight">d</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>Substitute the values of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span> into the cubic formula.</li> <li>Simplify the expression inside the cube root.</li> <li>Simplify the expression outside the cube root.</li> </ol> <h2><strong>Q: What are the real and complex solutions to the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>24</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^3 - 3x^2 - 10x + 24 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">24</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>?</strong></h2> <hr> <p>A: To find the real and complex solutions to the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>24</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^3 - 3x^2 - 10x + 24 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">24</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, we need to apply the cubic formula. After simplifying the expression, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mrow><mo>−</mo><mfrac><mn>5914</mn><mn>27</mn></mfrac><mo>+</mo><msqrt><mrow><mfrac><mn>34941196</mn><mn>729</mn></mfrac><mo>+</mo><mn>2197</mn></mrow></msqrt></mrow><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\frac{34941196}{729} + 2197}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.516em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.83em;"><span style="top:-2.9141em;"><span class="pstrut" style="height:3.6001em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.8301em;"><span class="pstrut" style="height:3.6001em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.83em;"><span class="pstrut" style="height:3.6001em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7783em;"><span style="top:-2.9561em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6001em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5914</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">34941196</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2197</span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span><span style="top:-3.5601em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478 c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514 c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20 s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121 s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081 l0 -0c4,-6.7,10,-10,18,-10 H400000 v40H1014.6 s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185 c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80 h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8399em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mrow><mo>−</mo><mfrac><mn>5914</mn><mn>27</mn></mfrac><mo>+</mo><msqrt><mfrac><mrow><mn>34941196</mn><mo>+</mo><mn>1598593</mn></mrow><mn>729</mn></mfrac></msqrt></mrow><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\frac{34941196 + 1598593}{729}}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.516em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.83em;"><span style="top:-2.9141em;"><span class="pstrut" style="height:3.6001em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.8301em;"><span class="pstrut" style="height:3.6001em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.83em;"><span class="pstrut" style="height:3.6001em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7783em;"><span style="top:-2.9561em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6001em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5914</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">34941196</span><span class="mbin mtight">+</span><span class="mord mtight">1598593</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span><span style="top:-3.5601em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478 c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514 c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20 s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121 s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081 l0 -0c4,-6.7,10,-10,18,-10 H400000 v40H1014.6 s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185 c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80 h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8399em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mrow><mo>−</mo><mfrac><mn>5914</mn><mn>27</mn></mfrac><mo>+</mo><msqrt><mfrac><mn>36539789</mn><mn>729</mn></mfrac></msqrt></mrow><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{-\frac{5914}{27} + \sqrt{\frac{36539789}{729}}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.516em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.83em;"><span style="top:-2.9141em;"><span class="pstrut" style="height:3.6001em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.8301em;"><span class="pstrut" style="height:3.6001em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.83em;"><span class="pstrut" style="height:3.6001em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7783em;"><span style="top:-2.9561em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6001em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5914</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">36539789</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span><span style="top:-3.5601em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478 c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514 c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20 s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121 s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081 l0 -0c4,-6.7,10,-10,18,-10 H400000 v40H1014.6 s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185 c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80 h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8399em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mfrac><mrow><mo>−</mo><mn>5914</mn><mo>⋅</mo><mn>27</mn><mo>+</mo><mn>36539789</mn></mrow><mn>729</mn></mfrac><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{\frac{-5914 \cdot 27 + 36539789}{729}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.916em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.23em;"><span style="top:-2.5491em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.4651em;"><span class="pstrut" style="height:3.2351em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.23em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7003em;"><span style="top:-2.8781em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">5914</span><span class="mbin mtight">⋅</span><span class="mord mtight">27</span><span class="mbin mtight">+</span><span class="mord mtight">36539789</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mfrac><mrow><mo>−</mo><mn>159414</mn><mo>−</mo><mn>36539789</mn></mrow><mn>729</mn></mfrac><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{\frac{-159414 - 36539789}{729}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.916em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.23em;"><span style="top:-2.5491em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.4651em;"><span class="pstrut" style="height:3.2351em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.23em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7003em;"><span style="top:-2.8781em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">159414</span><span class="mbin mtight">−</span><span class="mord mtight">36539789</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mfrac><mrow><mo>−</mo><mn>36540003</mn></mrow><mn>729</mn></mfrac><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{\frac{-36540003}{729}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.916em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.23em;"><span style="top:-2.5491em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.4651em;"><span class="pstrut" style="height:3.2351em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.23em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7003em;"><span style="top:-2.8781em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">36540003</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mo>±</mo><mroot><mrow><mo>−</mo><mfrac><mn>36540003</mn><mn>729</mn></mfrac></mrow><mn>3</mn></mroot></mrow><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">x = \frac{3 \pm \sqrt[3]{-\frac{36540003}{729}}}{3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.916em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.23em;"><span style="top:-2.5491em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.4651em;"><span class="pstrut" style="height:3.2351em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.23em;"><span class="pstrut" style="height:3.2351em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7003em;"><span style="top:-2.8781em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2351em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">729</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">36540003</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1951em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6049em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <h2><strong>Q: How do I find the real and complex solutions to a cubic equation?</strong></h2> <hr> <p>A: To find the real and complex solutions to a cubic equation, you need to apply the cubic formula. This involves calculating the values of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>, substituting them into the cubic formula, and simplifying the expression.</p> <h2><strong>Q: What are the advantages and disadvantages of using the cubic formula?</strong></h2> <hr> <p>A: The advantages of using the cubic formula include:</p> <ul> <li>It is a general method for solving cubic equations.</li> <li>It can be used to find both real and complex solutions.</li> <li>It is a powerful tool for solving cubic equations.</li> </ul> <p>The disadvantages of using the cubic formula include:</p> <ul> <li>It can be complex and difficult to apply.</li> <li>It requires a good understanding of algebra and calculus.</li> <li>It can be time-consuming to simplify the expression.</li> </ul> <h2><strong>Q: Can I use other methods to solve cubic equations?</strong></h2> <hr> <p>A: Yes, there are other methods to solve cubic equations, including:</p> <ul> <li>Factoring: This involves expressing the cubic equation as a product of linear factors.</li> <li>The Rational Root Theorem: This involves using the theorem to find possible rational roots of the cubic equation.</li> <li>Numerical methods: This involves using numerical methods, such as the Newton-Raphson method, to approximate the solutions to the cubic equation.</li> </ul> <h2><strong>Q: How do I choose the best method to solve a cubic equation?</strong></h2> <hr> <p>A: To choose the best method to solve a cubic equation, you need to consider the following factors:</p> <ul> <li>The complexity of the equation: If the equation is simple, factoring or the Rational Root Theorem may be sufficient. If the equation is complex, the cubic formula may be necessary.</li> <li>The desired level of accuracy: If you need an exact solution, the cubic formula may be necessary. If you are satisfied with an approximate solution, numerical methods may be sufficient.</li> <li>Your level of expertise: If you are not familiar with the cubic formula, you may want to start with simpler methods, such as factoring or the Rational Root Theorem.</li> </ul>