Solve Using the Square Root Property 13x^2+5=0

$13 x^{2} + 5 = 0$

Step 1

Subtract $5$ from both sides of the equation.

$13 x^{2} = - 5$

Step 2

Divide each term in $13 x^{2} = - 5$ by $13$ and simplify.

Divide each term in $13 x^{2} = - 5$ by $13$ .

$\frac{13 x^{2}}{13} = \frac{- 5}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 x^{2}}{13} = \frac{- 5}{13}$

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

$x^{2} = \frac{- 5}{13}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{5}{13}$

Step 3

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

$x = \pm \sqrt{- \frac{5}{13}}$

Step 4

Simplify $\pm \sqrt{- \frac{5}{13}}$ .

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

$x = \pm \sqrt{i^{2} \frac{5}{13}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{5}{13}}$

Rewrite $\sqrt{\frac{5}{13}}$ as $\frac{\sqrt{5}}{\sqrt{13}}$ .

$x = \pm i \frac{\sqrt{5}}{\sqrt{13}}$

Multiply $\frac{\sqrt{5}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \pm i (\frac{\sqrt{5}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{5}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}$

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

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{{\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

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

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

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

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{13^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{13^{1}}$

Evaluate the exponent.

$x = \pm i \frac{\sqrt{5} \sqrt{13}}{13}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{5 \cdot 13}}{13}$

Multiply $5$ by $13$ .

$x = \pm i \frac{\sqrt{65}}{13}$

Combine $i$ and $\frac{\sqrt{65}}{13}$ .

$x = \pm \frac{i \sqrt{65}}{13}$

Step 5

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

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

$x = \frac{i \sqrt{65}}{13}$

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

$x = - \frac{i \sqrt{65}}{13}$

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

$x = \frac{i \sqrt{65}}{13}, - \frac{i \sqrt{65}}{13}$

Solve Using the Square Root Property 13x^2+5=0

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