Solve 2sqrt{100^2}+sqrt{x^2}=300-x

$$2 \sqrt{ { 100 }^{ 2 } } + \sqrt{ { x }^{ 2 } } =300-x$$

Solve for x

$$x=50$$

Solution Steps

Subtract 2\sqrt{100^{2}} from both sides of the equation.

$$\sqrt{x^{2}}=300-x-2\sqrt{100^{2}}$$

Calculate 100 to the power of 2 and get 10000.

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

Calculate the square root of 10000 and get 100.

$$\sqrt{x^{2}}=300-x-2\times 100$$

Multiply -2 and 100 to get -200.

$$\sqrt{x^{2}}=300-x-200$$

Subtract 200 from 300 to get 100.

$$\sqrt{x^{2}}=100-x$$

Square both sides of the equation.

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

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

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

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

$$x^{2}=10000-200x+x^{2}$$

Add 200x to both sides.

$$x^{2}+200x=10000+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}+200x-x^{2}=10000$$

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

$$200x=10000$$

Divide both sides by 200.

$$x=\frac{10000}{200}$$

Divide 10000 by 200 to get 50.

$$x=50$$

Substitute 50 for x in the equation 2\sqrt{100^{2}}+\sqrt{x^{2}}=300-x.

$$2\sqrt{100^{2}}+\sqrt{50^{2}}=300-50$$

Simplify. The value x=50 satisfies the equation.

$$250=250$$

Equation \sqrt{x^{2}}=100-x has a unique solution.

$$x=50$$