Solve sqrt{30^2+x^2}=sqrt{20^2+(10-x)^2}30

$$\sqrt { 30 ^ { 2 } + x ^ { 2 } } = \sqrt { 20 ^ { 2 } + ( 10 - x ) ^ { 2 } } 30$$

Solve for x (complex solution)

$$x=\frac{-30\sqrt{358601}i+9000}{899}\approx 10.011123471-19.983304733i$$

$$x=\frac{9000+30\sqrt{358601}i}{899}\approx 10.011123471+19.983304733i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{30^{2}+x^{2}}\right)^{2}=\left(30\sqrt{20^{2}+\left(10-x\right)^{2}}\right)^{2}$$

Calculate 30 to the power of 2 and get 900.

$$\left(\sqrt{900+x^{2}}\right)^{2}=\left(30\sqrt{20^{2}+\left(10-x\right)^{2}}\right)^{2}$$

Calculate \sqrt{900+x^{2}} to the power of 2 and get 900+x^{2}.

$$900+x^{2}=\left(30\sqrt{20^{2}+\left(10-x\right)^{2}}\right)^{2}$$

Calculate 20 to the power of 2 and get 400.

$$900+x^{2}=\left(30\sqrt{400+\left(10-x\right)^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-x\right)^{2}.

$$900+x^{2}=\left(30\sqrt{400+100-20x+x^{2}}\right)^{2}$$

Add 400 and 100 to get 500.

$$900+x^{2}=\left(30\sqrt{500-20x+x^{2}}\right)^{2}$$

Expand \left(30\sqrt{500-20x+x^{2}}\right)^{2}.

$$900+x^{2}=30^{2}\left(\sqrt{500-20x+x^{2}}\right)^{2}$$

Calculate 30 to the power of 2 and get 900.

$$900+x^{2}=900\left(\sqrt{500-20x+x^{2}}\right)^{2}$$

Calculate \sqrt{500-20x+x^{2}} to the power of 2 and get 500-20x+x^{2}.

$$900+x^{2}=900\left(500-20x+x^{2}\right)$$

Use the distributive property to multiply 900 by 500-20x+x^{2}.

$$900+x^{2}=450000-18000x+900x^{2}$$

Subtract 450000 from both sides.

$$900+x^{2}-450000=-18000x+900x^{2}$$

Subtract 450000 from 900 to get -449100.

$$-449100+x^{2}=-18000x+900x^{2}$$

Add 18000x to both sides.

$$-449100+x^{2}+18000x=900x^{2}$$

Subtract 900x^{2} from both sides.

$$-449100+x^{2}+18000x-900x^{2}=0$$

Combine x^{2} and -900x^{2} to get -899x^{2}.

$$-449100-899x^{2}+18000x=0$$

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.

$$-899x^{2}+18000x-449100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -899 for a, 18000 for b, and -449100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18000±\sqrt{18000^{2}-4\left(-899\right)\left(-449100\right)}}{2\left(-899\right)}$$

Square 18000.

$$x=\frac{-18000±\sqrt{324000000-4\left(-899\right)\left(-449100\right)}}{2\left(-899\right)}$$

Multiply -4 times -899.

$$x=\frac{-18000±\sqrt{324000000+3596\left(-449100\right)}}{2\left(-899\right)}$$

Multiply 3596 times -449100.

$$x=\frac{-18000±\sqrt{324000000-1614963600}}{2\left(-899\right)}$$

Add 324000000 to -1614963600.

$$x=\frac{-18000±\sqrt{-1290963600}}{2\left(-899\right)}$$

Take the square root of -1290963600.

$$x=\frac{-18000±60\sqrt{358601}i}{2\left(-899\right)}$$

Multiply 2 times -899.

$$x=\frac{-18000±60\sqrt{358601}i}{-1798}$$

Now solve the equation x=\frac{-18000±60\sqrt{358601}i}{-1798} when ± is plus. Add -18000 to 60i\sqrt{358601}.

$$x=\frac{-18000+60\sqrt{358601}i}{-1798}$$

Divide -18000+60i\sqrt{358601} by -1798.

$$x=\frac{-30\sqrt{358601}i+9000}{899}$$

Now solve the equation x=\frac{-18000±60\sqrt{358601}i}{-1798} when ± is minus. Subtract 60i\sqrt{358601} from -18000.

$$x=\frac{-60\sqrt{358601}i-18000}{-1798}$$

Divide -18000-60i\sqrt{358601} by -1798.

$$x=\frac{9000+30\sqrt{358601}i}{899}$$

The equation is now solved.

$$x=\frac{-30\sqrt{358601}i+9000}{899} x=\frac{9000+30\sqrt{358601}i}{899}$$

Substitute \frac{-30\sqrt{358601}i+9000}{899} for x in the equation \sqrt{30^{2}+x^{2}}=30\sqrt{20^{2}+\left(10-x\right)^{2}}.

$$\sqrt{30^{2}+\left(\frac{-30\sqrt{358601}i+9000}{899}\right)^{2}}=30\sqrt{20^{2}+\left(10-\frac{-30\sqrt{358601}i+9000}{899}\right)^{2}}$$

Simplify. The value x=\frac{-30\sqrt{358601}i+9000}{899} satisfies the equation.

$$\frac{30}{899}\left(539600-600i\times 358601^{\frac{1}{2}}\right)^{\frac{1}{2}}=\frac{300}{899}\left(5396-6i\times 358601^{\frac{1}{2}}\right)^{\frac{1}{2}}$$

Substitute \frac{9000+30\sqrt{358601}i}{899} for x in the equation \sqrt{30^{2}+x^{2}}=30\sqrt{20^{2}+\left(10-x\right)^{2}}.

$$\sqrt{30^{2}+\left(\frac{9000+30\sqrt{358601}i}{899}\right)^{2}}=30\sqrt{20^{2}+\left(10-\frac{9000+30\sqrt{358601}i}{899}\right)^{2}}$$

Simplify. The value x=\frac{9000+30\sqrt{358601}i}{899} satisfies the equation.

$$\frac{30}{899}\left(539600+600i\times 358601^{\frac{1}{2}}\right)^{\frac{1}{2}}=\frac{300}{899}\left(5396+6i\times 358601^{\frac{1}{2}}\right)^{\frac{1}{2}}$$

List all solutions of \sqrt{x^{2}+900}=30\sqrt{\left(10-x\right)^{2}+400}.

$$x=\frac{-30\sqrt{358601}i+9000}{899} x=\frac{9000+30\sqrt{358601}i}{899}$$