Solve 80=x+sqrt{36+x^2}

$$80 = x + \sqrt { 36 + x ^ { 2 } }$$

Solve for x

$$x = \frac{1591}{40} = 39\frac{31}{40} = 39.775$$

Solution Steps

Subtract x from both sides of the equation.

$$80-x=\sqrt{36+x^{2}}$$

Square both sides of the equation.

$$\left(80-x\right)^{2}=\left(\sqrt{36+x^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(80-x\right)^{2}.

$$6400-160x+x^{2}=\left(\sqrt{36+x^{2}}\right)^{2}$$

Calculate \sqrt{36+x^{2}} to the power of 2 and get 36+x^{2}.

$$6400-160x+x^{2}=36+x^{2}$$

Subtract x^{2} from both sides.

$$6400-160x+x^{2}-x^{2}=36$$

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

$$6400-160x=36$$

Subtract 6400 from both sides.

$$-160x=36-6400$$

Subtract 6400 from 36 to get -6364.

$$-160x=-6364$$

Divide both sides by -160.

$$x=\frac{-6364}{-160}$$

Reduce the fraction \frac{-6364}{-160} to lowest terms by extracting and canceling out -4.

$$x=\frac{1591}{40}$$

Substitute \frac{1591}{40} for x in the equation 80=x+\sqrt{36+x^{2}}.

$$80=\frac{1591}{40}+\sqrt{36+\left(\frac{1591}{40}\right)^{2}}$$

Simplify. The value x=\frac{1591}{40} satisfies the equation.

$$80=80$$

Equation 80-x=\sqrt{x^{2}+36} has a unique solution.

$$x=\frac{1591}{40}$$