Solve 200+x=2sqrt{(100-x)^2+100^2}

$$200 + x = 2 \sqrt { ( 100 - x ) ^ { 2 } + 100 ^ { 2 } }$$

Solve for x

$$x=\frac{200\sqrt{6}}{3}+200\approx 363.299316186$$

$$x=-\frac{200\sqrt{6}}{3}+200\approx 36.700683814$$

Solution Steps

Square both sides of the equation.

$$\left(200+x\right)^{2}=\left(2\sqrt{\left(100-x\right)^{2}+100^{2}}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(200+x\right)^{2}.

$$40000+400x+x^{2}=\left(2\sqrt{\left(100-x\right)^{2}+100^{2}}\right)^{2}$$

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

$$40000+400x+x^{2}=\left(2\sqrt{10000-200x+x^{2}+100^{2}}\right)^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$40000+400x+x^{2}=\left(2\sqrt{10000-200x+x^{2}+10000}\right)^{2}$$

Add 10000 and 10000 to get 20000.

$$40000+400x+x^{2}=\left(2\sqrt{20000-200x+x^{2}}\right)^{2}$$

Expand \left(2\sqrt{20000-200x+x^{2}}\right)^{2}.

$$40000+400x+x^{2}=2^{2}\left(\sqrt{20000-200x+x^{2}}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$40000+400x+x^{2}=4\left(\sqrt{20000-200x+x^{2}}\right)^{2}$$

Calculate \sqrt{20000-200x+x^{2}} to the power of 2 and get 20000-200x+x^{2}.

$$40000+400x+x^{2}=4\left(20000-200x+x^{2}\right)$$

Use the distributive property to multiply 4 by 20000-200x+x^{2}.

$$40000+400x+x^{2}=80000-800x+4x^{2}$$

Subtract 80000 from both sides.

$$40000+400x+x^{2}-80000=-800x+4x^{2}$$

Subtract 80000 from 40000 to get -40000.

$$-40000+400x+x^{2}=-800x+4x^{2}$$

Add 800x to both sides.

$$-40000+400x+x^{2}+800x=4x^{2}$$

Combine 400x and 800x to get 1200x.

$$-40000+1200x+x^{2}=4x^{2}$$

Subtract 4x^{2} from both sides.

$$-40000+1200x+x^{2}-4x^{2}=0$$

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

$$-40000+1200x-3x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-3x^{2}+1200x-40000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 1200 for b, and -40000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-3\right)\left(-40000\right)}}{2\left(-3\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-3\right)\left(-40000\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-1200±\sqrt{1440000+12\left(-40000\right)}}{2\left(-3\right)}$$

Multiply 12 times -40000.

$$x=\frac{-1200±\sqrt{1440000-480000}}{2\left(-3\right)}$$

Add 1440000 to -480000.

$$x=\frac{-1200±\sqrt{960000}}{2\left(-3\right)}$$

Take the square root of 960000.

$$x=\frac{-1200±400\sqrt{6}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-1200±400\sqrt{6}}{-6}$$

Now solve the equation x=\frac{-1200±400\sqrt{6}}{-6} when ± is plus. Add -1200 to 400\sqrt{6}.

$$x=\frac{400\sqrt{6}-1200}{-6}$$

Divide -1200+400\sqrt{6} by -6.

$$x=-\frac{200\sqrt{6}}{3}+200$$

Now solve the equation x=\frac{-1200±400\sqrt{6}}{-6} when ± is minus. Subtract 400\sqrt{6} from -1200.

$$x=\frac{-400\sqrt{6}-1200}{-6}$$

Divide -1200-400\sqrt{6} by -6.

$$x=\frac{200\sqrt{6}}{3}+200$$

The equation is now solved.

$$x=-\frac{200\sqrt{6}}{3}+200 x=\frac{200\sqrt{6}}{3}+200$$

Substitute -\frac{200\sqrt{6}}{3}+200 for x in the equation 200+x=2\sqrt{\left(100-x\right)^{2}+100^{2}}.

$$200-\frac{200\sqrt{6}}{3}+200=2\sqrt{\left(100-\left(-\frac{200\sqrt{6}}{3}+200\right)\right)^{2}+100^{2}}$$

Simplify. The value x=-\frac{200\sqrt{6}}{3}+200 satisfies the equation.

$$400-\frac{200}{3}\times 6^{\frac{1}{2}}=400-\frac{200}{3}\times 6^{\frac{1}{2}}$$

Substitute \frac{200\sqrt{6}}{3}+200 for x in the equation 200+x=2\sqrt{\left(100-x\right)^{2}+100^{2}}.

$$200+\frac{200\sqrt{6}}{3}+200=2\sqrt{\left(100-\left(\frac{200\sqrt{6}}{3}+200\right)\right)^{2}+100^{2}}$$

Simplify. The value x=\frac{200\sqrt{6}}{3}+200 satisfies the equation.

$$400+\frac{200}{3}\times 6^{\frac{1}{2}}=400+\frac{200}{3}\times 6^{\frac{1}{2}}$$

List all solutions of x+200=2\sqrt{\left(100-x\right)^{2}+10000}.

$$x=-\frac{200\sqrt{6}}{3}+200 x=\frac{200\sqrt{6}}{3}+200$$