Solve Using the Square Root Property 4x^2-13=0

$4 x^{2} - 13 = 0$

Step 1

Add $13$ to both sides of the equation.

$4 x^{2} = 13$

Step 2

Divide each term in $4 x^{2} = 13$ by $4$ and simplify.

Divide each term in $4 x^{2} = 13$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{13}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{13}{4}$

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

$x^{2} = \frac{13}{4}$

Step 3

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

$x = \pm \sqrt{\frac{13}{4}}$

Step 4

Simplify $\pm \sqrt{\frac{13}{4}}$ .

Rewrite $\sqrt{\frac{13}{4}}$ as $\frac{\sqrt{13}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{13}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

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.

$x = \frac{\sqrt{13}}{2}$

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

$x = - \frac{\sqrt{13}}{2}$

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

$x = \frac{\sqrt{13}}{2}, - \frac{\sqrt{13}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{13}}{2}, - \frac{\sqrt{13}}{2}$

Decimal Form:

$x = 1.80277563 \dots, - 1.80277563 \dots$