Finite Math Examples

$24 x^{2} + 13 = 0$

Step 1

Subtract $13$ from both sides of the equation.

$24 x^{2} = - 13$

Step 2

Divide each term in $24 x^{2} = - 13$ by $24$ and simplify.

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Step 2.1

Divide each term in $24 x^{2} = - 13$ by $24$ .

$\frac{24 x^{2}}{24} = \frac{- 13}{24}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $24$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{24 x^{2}}{24} = \frac{- 13}{24}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 13}{24}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{13}{24}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{13}{24}}$

Step 4

Simplify $\pm \sqrt{- \frac{13}{24}}$ .

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Step 4.1

Rewrite $- \frac{13}{24}$ as ${(\frac{i}{2})}^{2} \frac{13}{6}$ .

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Step 4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{13}{24}}$

Step 4.1.2

Factor the perfect power $1^{2}$ out of $13$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 13}{24}}$

Step 4.1.3

Factor the perfect power $2^{2}$ out of $24$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 13}{2^{2} \cdot 6}}$

Step 4.1.4

Rearrange the fraction $\frac{1^{2} \cdot 13}{2^{2} \cdot 6}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \frac{13}{6})}$

Step 4.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2} \frac{13}{6}}$

Step 4.2

Pull terms out from under the radical.

$x = \pm \frac{i}{2} \sqrt{\frac{13}{6}}$

Step 4.3

Rewrite $\sqrt{\frac{13}{6}}$ as $\frac{\sqrt{13}}{\sqrt{6}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13}}{\sqrt{6}}$

Step 4.4

Multiply $\frac{\sqrt{13}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{i}{2} (\frac{\sqrt{13}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{\sqrt{13}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 4.5.2

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 4.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 4.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{6^{1}}$

Step 4.5.6.5

Evaluate the exponent.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13} \sqrt{6}}{6}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Combine using the product rule for radicals.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{13 \cdot 6}}{6}$

Step 4.6.2

Multiply $13$ by $6$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{78}}{6}$

Step 4.7

Multiply $\frac{i}{2} \cdot \frac{\sqrt{78}}{6}$ .

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Step 4.7.1

Multiply $\frac{i}{2}$ by $\frac{\sqrt{78}}{6}$ .

$x = \pm \frac{i \sqrt{78}}{2 \cdot 6}$

Step 4.7.2

Multiply $2$ by $6$ .

$x = \pm \frac{i \sqrt{78}}{12}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{78}}{12}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{78}}{12}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{78}}{12}, - \frac{i \sqrt{78}}{12}$