Solve for x square root of x^2-7=-2

$\sqrt{x^{2} - 7} = - 2$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} - 7}}^{2} = {(- 2)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 7}$ as ${(x^{2} - 7)}^{\frac{1}{2}}$ .

${({(x^{2} - 7)}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} - 7)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} - 7)}^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 7)}^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Rewrite the expression.

${(x^{2} - 7)}^{1} = {(- 2)}^{2}$

Simplify.

$x^{2} - 7 = {(- 2)}^{2}$

Simplify the right side.

Raise $- 2$ to the power of $2$ .

$x^{2} - 7 = 4$

Step 3

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Add $7$ to both sides of the equation.

$x^{2} = 4 + 7$

Add $4$ and $7$ .

$x^{2} = 11$

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

$x = \pm \sqrt{11}$

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.

$x = \sqrt{11}$

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

$x = - \sqrt{11}$

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

$x = \sqrt{11}, - \sqrt{11}$

Step 4

Exclude the solutions that do not make $\sqrt{x^{2} - 7} = - 2$ true.

$No solution$