Solve x^2+left(30-xright)^2=left(13^2right)^

$${ x }^{ 2 } + { \left(30-x \right) }^{ 2 } = { \left( { 13 }^{ 2 } \right) }^{ }$$

Solve for x (complex solution)

$$x=\frac{\sqrt{562}i}{2}+15\approx 15+11.853269591i$$

$$x=-\frac{\sqrt{562}i}{2}+15\approx 15-11.853269591i$$

Steps Using the Quadratic Formula

To raise a power to another power, multiply the exponents. Multiply 2 and 1 to get 2.

$$x^{2}+\left(30-x\right)^{2}=13^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(30-x\right)^{2}.

$$x^{2}+900-60x+x^{2}=13^{2}$$

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

$$2x^{2}+900-60x=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$2x^{2}+900-60x=169$$

Subtract 169 from both sides.

$$2x^{2}+900-60x-169=0$$

Subtract 169 from 900 to get 731.

$$2x^{2}+731-60x=0$$

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.

$$2x^{2}-60x+731=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -60 for b, and 731 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 2\times 731}}{2\times 2}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 2\times 731}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-60\right)±\sqrt{3600-8\times 731}}{2\times 2}$$

Multiply -8 times 731.

$$x=\frac{-\left(-60\right)±\sqrt{3600-5848}}{2\times 2}$$

Add 3600 to -5848.

$$x=\frac{-\left(-60\right)±\sqrt{-2248}}{2\times 2}$$

Take the square root of -2248.

$$x=\frac{-\left(-60\right)±2\sqrt{562}i}{2\times 2}$$

The opposite of -60 is 60.

$$x=\frac{60±2\sqrt{562}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{60±2\sqrt{562}i}{4}$$

Now solve the equation x=\frac{60±2\sqrt{562}i}{4} when ± is plus. Add 60 to 2i\sqrt{562}.

$$x=\frac{60+2\sqrt{562}i}{4}$$

Divide 60+2i\sqrt{562} by 4.

$$x=\frac{\sqrt{562}i}{2}+15$$

Now solve the equation x=\frac{60±2\sqrt{562}i}{4} when ± is minus. Subtract 2i\sqrt{562} from 60.

$$x=\frac{-2\sqrt{562}i+60}{4}$$

Divide 60-2i\sqrt{562} by 4.

$$x=-\frac{\sqrt{562}i}{2}+15$$

The equation is now solved.

$$x=\frac{\sqrt{562}i}{2}+15 x=-\frac{\sqrt{562}i}{2}+15$$

Steps for Completing the Square

To raise a power to another power, multiply the exponents. Multiply 2 and 1 to get 2.

$$x^{2}+\left(30-x\right)^{2}=13^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(30-x\right)^{2}.

$$x^{2}+900-60x+x^{2}=13^{2}$$

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

$$2x^{2}+900-60x=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$2x^{2}+900-60x=169$$

Subtract 900 from both sides.

$$2x^{2}-60x=169-900$$

Subtract 900 from 169 to get -731.

$$2x^{2}-60x=-731$$

Divide both sides by 2.

$$\frac{2x^{2}-60x}{2}=-\frac{731}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{60}{2}\right)x=-\frac{731}{2}$$

Divide -60 by 2.

$$x^{2}-30x=-\frac{731}{2}$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-\frac{731}{2}+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-\frac{731}{2}+225$$

Add -\frac{731}{2} to 225.

$$x^{2}-30x+225=-\frac{281}{2}$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=-\frac{281}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{-\frac{281}{2}}$$

Simplify.

$$x-15=\frac{\sqrt{562}i}{2} x-15=-\frac{\sqrt{562}i}{2}$$

Add 15 to both sides of the equation.

$$x=\frac{\sqrt{562}i}{2}+15 x=-\frac{\sqrt{562}i}{2}+15$$