Solve (100)^2+x^2=(2x+100)^2

$$( 100 ) ^ { 2 } + x ^ { 2 } = ( 2 x + 100 ) ^ { 2 }$$

Solve for x

$$x = -\frac{400}{3} = -133\frac{1}{3} \approx -133.333333333$$

$$x=0$$

Steps Using Factoring

Calculate 100 to the power of 2 and get 10000.

$$10000+x^{2}=\left(2x+100\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+100\right)^{2}.

$$10000+x^{2}=4x^{2}+400x+10000$$

Subtract 4x^{2} from both sides.

$$10000+x^{2}-4x^{2}=400x+10000$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$10000-3x^{2}=400x+10000$$

Subtract 400x from both sides.

$$10000-3x^{2}-400x=10000$$

Subtract 10000 from both sides.

$$10000-3x^{2}-400x-10000=0$$

Subtract 10000 from 10000 to get 0.

$$-3x^{2}-400x=0$$

Factor out x.

$$x\left(-3x-400\right)=0$$

To find equation solutions, solve x=0 and -3x-400=0.

$$x=0 x=-\frac{400}{3}$$

Steps Using the Quadratic Formula

Calculate 100 to the power of 2 and get 10000.

$$10000+x^{2}=\left(2x+100\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+100\right)^{2}.

$$10000+x^{2}=4x^{2}+400x+10000$$

Subtract 4x^{2} from both sides.

$$10000+x^{2}-4x^{2}=400x+10000$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$10000-3x^{2}=400x+10000$$

Subtract 400x from both sides.

$$10000-3x^{2}-400x=10000$$

Subtract 10000 from both sides.

$$10000-3x^{2}-400x-10000=0$$

Subtract 10000 from 10000 to get 0.

$$-3x^{2}-400x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -400 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}}}{2\left(-3\right)}$$

Take the square root of \left(-400\right)^{2}.

$$x=\frac{-\left(-400\right)±400}{2\left(-3\right)}$$

The opposite of -400 is 400.

$$x=\frac{400±400}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{400±400}{-6}$$

Now solve the equation x=\frac{400±400}{-6} when ± is plus. Add 400 to 400.

$$x=\frac{800}{-6}$$

Reduce the fraction \frac{800}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{400}{3}$$

Now solve the equation x=\frac{400±400}{-6} when ± is minus. Subtract 400 from 400.

$$x=\frac{0}{-6}$$

Divide 0 by -6.

$$x=0$$

The equation is now solved.

$$x=-\frac{400}{3} x=0$$

Steps for Completing the Square

Calculate 100 to the power of 2 and get 10000.

$$10000+x^{2}=\left(2x+100\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+100\right)^{2}.

$$10000+x^{2}=4x^{2}+400x+10000$$

Subtract 4x^{2} from both sides.

$$10000+x^{2}-4x^{2}=400x+10000$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$10000-3x^{2}=400x+10000$$

Subtract 400x from both sides.

$$10000-3x^{2}-400x=10000$$

Subtract 10000 from both sides.

$$-3x^{2}-400x=10000-10000$$

Subtract 10000 from 10000 to get 0.

$$-3x^{2}-400x=0$$

Divide both sides by -3.

$$\frac{-3x^{2}-400x}{-3}=\frac{0}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{400}{-3}\right)x=\frac{0}{-3}$$

Divide -400 by -3.

$$x^{2}+\frac{400}{3}x=\frac{0}{-3}$$

Divide 0 by -3.

$$x^{2}+\frac{400}{3}x=0$$

Divide \frac{400}{3}, the coefficient of the x term, by 2 to get \frac{200}{3}. Then add the square of \frac{200}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{400}{3}x+\left(\frac{200}{3}\right)^{2}=\left(\frac{200}{3}\right)^{2}$$

Square \frac{200}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{400}{3}x+\frac{40000}{9}=\frac{40000}{9}$$

Factor x^{2}+\frac{400}{3}x+\frac{40000}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{200}{3}\right)^{2}=\frac{40000}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{200}{3}\right)^{2}}=\sqrt{\frac{40000}{9}}$$

Simplify.

$$x+\frac{200}{3}=\frac{200}{3} x+\frac{200}{3}=-\frac{200}{3}$$

Subtract \frac{200}{3} from both sides of the equation.

$$x=0 x=-\frac{400}{3}$$