Solve sqrt{x^2-14x+113-9^2}=x

$$\sqrt{ { x }^{ 2 } -14x+113- { 9 }^{ 2 } } =x$$

Solve for x

$$x = \frac{16}{7} = 2\frac{2}{7} \approx 2.285714286$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x^{2}-14x+113-9^{2}}\right)^{2}=x^{2}$$

Calculate 9 to the power of 2 and get 81.

$$\left(\sqrt{x^{2}-14x+113-81}\right)^{2}=x^{2}$$

Subtract 81 from 113 to get 32.

$$\left(\sqrt{x^{2}-14x+32}\right)^{2}=x^{2}$$

Calculate \sqrt{x^{2}-14x+32} to the power of 2 and get x^{2}-14x+32.

$$x^{2}-14x+32=x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}-14x+32-x^{2}=0$$

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

$$-14x+32=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$-14x=-32$$

Divide both sides by -14.

$$x=\frac{-32}{-14}$$

Reduce the fraction \frac{-32}{-14} to lowest terms by extracting and canceling out -2.

$$x=\frac{16}{7}$$

Substitute \frac{16}{7} for x in the equation \sqrt{x^{2}-14x+113-9^{2}}=x.

$$\sqrt{\left(\frac{16}{7}\right)^{2}-14\times \frac{16}{7}+113-9^{2}}=\frac{16}{7}$$

Simplify. The value x=\frac{16}{7} satisfies the equation.

$$\frac{16}{7}=\frac{16}{7}$$

Equation \sqrt{x^{2}-14x+113-81}=x has a unique solution.

$$x=\frac{16}{7}$$