Solve 300-x=2sqrt{10000+x^2}

$$300-x = 2 \sqrt{ 10000+ { x }^{ 2 } }$$

Solve for x

$$x=\frac{200\sqrt{6}}{3}-100\approx 63.299316186$$

$$x=-\frac{200\sqrt{6}}{3}-100\approx -263.299316186$$

Solution Steps

Square both sides of the equation.

$$\left(300-x\right)^{2}=\left(2\sqrt{10000+x^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(300-x\right)^{2}.

$$90000-600x+x^{2}=\left(2\sqrt{10000+x^{2}}\right)^{2}$$

Expand \left(2\sqrt{10000+x^{2}}\right)^{2}.

$$90000-600x+x^{2}=2^{2}\left(\sqrt{10000+x^{2}}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$90000-600x+x^{2}=4\left(\sqrt{10000+x^{2}}\right)^{2}$$

Calculate \sqrt{10000+x^{2}} to the power of 2 and get 10000+x^{2}.

$$90000-600x+x^{2}=4\left(10000+x^{2}\right)$$

Use the distributive property to multiply 4 by 10000+x^{2}.

$$90000-600x+x^{2}=40000+4x^{2}$$

Subtract 40000 from both sides.

$$90000-600x+x^{2}-40000=4x^{2}$$

Subtract 40000 from 90000 to get 50000.

$$50000-600x+x^{2}=4x^{2}$$

Subtract 4x^{2} from both sides.

$$50000-600x+x^{2}-4x^{2}=0$$

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

$$50000-600x-3x^{2}=0$$

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.

$$-3x^{2}-600x+50000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -600 for b, and 50000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-600\right)±\sqrt{\left(-600\right)^{2}-4\left(-3\right)\times 50000}}{2\left(-3\right)}$$

Square -600.

$$x=\frac{-\left(-600\right)±\sqrt{360000-4\left(-3\right)\times 50000}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-600\right)±\sqrt{360000+12\times 50000}}{2\left(-3\right)}$$

Multiply 12 times 50000.

$$x=\frac{-\left(-600\right)±\sqrt{360000+600000}}{2\left(-3\right)}$$

Add 360000 to 600000.

$$x=\frac{-\left(-600\right)±\sqrt{960000}}{2\left(-3\right)}$$

Take the square root of 960000.

$$x=\frac{-\left(-600\right)±400\sqrt{6}}{2\left(-3\right)}$$

The opposite of -600 is 600.

$$x=\frac{600±400\sqrt{6}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{600±400\sqrt{6}}{-6}$$

Now solve the equation x=\frac{600±400\sqrt{6}}{-6} when ± is plus. Add 600 to 400\sqrt{6}.

$$x=\frac{400\sqrt{6}+600}{-6}$$

Divide 600+400\sqrt{6} by -6.

$$x=-\frac{200\sqrt{6}}{3}-100$$

Now solve the equation x=\frac{600±400\sqrt{6}}{-6} when ± is minus. Subtract 400\sqrt{6} from 600.

$$x=\frac{600-400\sqrt{6}}{-6}$$

Divide 600-400\sqrt{6} by -6.

$$x=\frac{200\sqrt{6}}{3}-100$$

The equation is now solved.

$$x=-\frac{200\sqrt{6}}{3}-100 x=\frac{200\sqrt{6}}{3}-100$$

Substitute -\frac{200\sqrt{6}}{3}-100 for x in the equation 300-x=2\sqrt{10000+x^{2}}.

$$300-\left(-\frac{200\sqrt{6}}{3}-100\right)=2\sqrt{10000+\left(-\frac{200\sqrt{6}}{3}-100\right)^{2}}$$

Simplify. The value x=-\frac{200\sqrt{6}}{3}-100 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}-100 for x in the equation 300-x=2\sqrt{10000+x^{2}}.

$$300-\left(\frac{200\sqrt{6}}{3}-100\right)=2\sqrt{10000+\left(\frac{200\sqrt{6}}{3}-100\right)^{2}}$$

Simplify. The value x=\frac{200\sqrt{6}}{3}-100 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 300-x=2\sqrt{x^{2}+10000}.

$$x=-\frac{200\sqrt{6}}{3}-100 x=\frac{200\sqrt{6}}{3}-100$$