Solve x+12=8sqrt{x}

$$x + 12 = 8 \sqrt { x }$$

Solve for x

$$x=36$$

$$x=4$$

Solution Steps

Square both sides of the equation.

$$\left(x+12\right)^{2}=\left(8\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$x^{2}+24x+144=\left(8\sqrt{x}\right)^{2}$$

Expand \left(8\sqrt{x}\right)^{2}.

$$x^{2}+24x+144=8^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$x^{2}+24x+144=64\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}+24x+144=64x$$

Subtract 64x from both sides.

$$x^{2}+24x+144-64x=0$$

Combine 24x and -64x to get -40x.

$$x^{2}-40x+144=0$$

To solve the equation, factor x^{2}-40x+144 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.

$$a+b=-40 ab=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -40.

$$a=-36 b=-4$$

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

$$\left(x-36\right)\left(x-4\right)$$

To find equation solutions, solve x-36=0 and x-4=0.

$$x=36 x=4$$

Substitute 36 for x in the equation x+12=8\sqrt{x}.

$$36+12=8\sqrt{36}$$

Simplify. The value x=36 satisfies the equation.

$$48=48$$

Substitute 4 for x in the equation x+12=8\sqrt{x}.

$$4+12=8\sqrt{4}$$

Simplify. The value x=4 satisfies the equation.

$$16=16$$

List all solutions of x+12=8\sqrt{x}.

$$x=36 x=4$$