Solve sqrt{left(12+xright)^2-16^2}=x

$$\sqrt{ { \left(12+x \right) }^{ 2 } - { 16 }^{ 2 } } =x$$

Solve for x

$$x = \frac{14}{3} = 4\frac{2}{3} \approx 4.666666667$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\left(12+x\right)^{2}-16^{2}}\right)^{2}=x^{2}$$

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

$$\left(\sqrt{144+24x+x^{2}-16^{2}}\right)^{2}=x^{2}$$

Calculate 16 to the power of 2 and get 256.

$$\left(\sqrt{144+24x+x^{2}-256}\right)^{2}=x^{2}$$

Subtract 256 from 144 to get -112.

$$\left(\sqrt{-112+24x+x^{2}}\right)^{2}=x^{2}$$

Calculate \sqrt{-112+24x+x^{2}} to the power of 2 and get -112+24x+x^{2}.

$$-112+24x+x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$-112+24x+x^{2}-x^{2}=0$$

Combine x^{2} and -x^{2} to get 0.

$$-112+24x=0$$

Add 112 to both sides. Anything plus zero gives itself.

$$24x=112$$

Divide both sides by 24.

$$x=\frac{112}{24}$$

Reduce the fraction \frac{112}{24} to lowest terms by extracting and canceling out 8.

$$x=\frac{14}{3}$$

Substitute \frac{14}{3} for x in the equation \sqrt{\left(12+x\right)^{2}-16^{2}}=x.

$$\sqrt{\left(12+\frac{14}{3}\right)^{2}-16^{2}}=\frac{14}{3}$$

Simplify. The value x=\frac{14}{3} satisfies the equation.

$$\frac{14}{3}=\frac{14}{3}$$

Equation \sqrt{\left(x+12\right)^{2}-256}=x has a unique solution.

$$x=\frac{14}{3}$$