Solve sqrt{x^2}=sqrt{4x+21}

$$\sqrt { x ^ { 2 } } = \sqrt { 4 x + 21 }$$

Solve for x

$$x=-3$$

$$x=7$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x^{2}}\right)^{2}=\left(\sqrt{4x+21}\right)^{2}$$

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

$$x^{2}=\left(\sqrt{4x+21}\right)^{2}$$

Calculate \sqrt{4x+21} to the power of 2 and get 4x+21.

$$x^{2}=4x+21$$

Subtract 4x from both sides.

$$x^{2}-4x=21$$

Subtract 21 from both sides.

$$x^{2}-4x-21=0$$

To solve the equation, factor x^{2}-4x-21 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -21.

$$1,-21 3,-7$$

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=-7 b=3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-7\right)\left(x+3\right)$$

To find equation solutions, solve x-7=0 and x+3=0.

$$x=7 x=-3$$

Substitute 7 for x in the equation \sqrt{x^{2}}=\sqrt{4x+21}.

$$\sqrt{7^{2}}=\sqrt{4\times 7+21}$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

Substitute -3 for x in the equation \sqrt{x^{2}}=\sqrt{4x+21}.

$$\sqrt{\left(-3\right)^{2}}=\sqrt{4\left(-3\right)+21}$$

Simplify. The value x=-3 satisfies the equation.

$$3=3$$

List all solutions of \sqrt{x^{2}}=\sqrt{4x+21}.

$$x=7 x=-3$$