Solve x^2+(x+395)^2=780

$$x ^ { 2 } + ( x + 395 ) ^ { 2 } = 780$$

Solve for x (complex solution)

$$x=\frac{-395+\sqrt{154465}i}{2}\approx -197.5+196.510177853i$$

$$x=\frac{-\sqrt{154465}i-395}{2}\approx -197.5-196.510177853i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+395\right)^{2}.

$$x^{2}+x^{2}+790x+156025=780$$

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

$$2x^{2}+790x+156025=780$$

Subtract 780 from both sides.

$$2x^{2}+790x+156025-780=0$$

Subtract 780 from 156025 to get 155245.

$$2x^{2}+790x+155245=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 790 for b, and 155245 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-790±\sqrt{790^{2}-4\times 2\times 155245}}{2\times 2}$$

Square 790.

$$x=\frac{-790±\sqrt{624100-4\times 2\times 155245}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-790±\sqrt{624100-8\times 155245}}{2\times 2}$$

Multiply -8 times 155245.

$$x=\frac{-790±\sqrt{624100-1241960}}{2\times 2}$$

Add 624100 to -1241960.

$$x=\frac{-790±\sqrt{-617860}}{2\times 2}$$

Take the square root of -617860.

$$x=\frac{-790±2\sqrt{154465}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-790±2\sqrt{154465}i}{4}$$

Now solve the equation x=\frac{-790±2\sqrt{154465}i}{4} when ± is plus. Add -790 to 2i\sqrt{154465}.

$$x=\frac{-790+2\sqrt{154465}i}{4}$$

Divide -790+2i\sqrt{154465} by 4.

$$x=\frac{-395+\sqrt{154465}i}{2}$$

Now solve the equation x=\frac{-790±2\sqrt{154465}i}{4} when ± is minus. Subtract 2i\sqrt{154465} from -790.

$$x=\frac{-2\sqrt{154465}i-790}{4}$$

Divide -790-2i\sqrt{154465} by 4.

$$x=\frac{-\sqrt{154465}i-395}{2}$$

The equation is now solved.

$$x=\frac{-395+\sqrt{154465}i}{2} x=\frac{-\sqrt{154465}i-395}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+395\right)^{2}.

$$x^{2}+x^{2}+790x+156025=780$$

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

$$2x^{2}+790x+156025=780$$

Subtract 156025 from both sides.

$$2x^{2}+790x=780-156025$$

Subtract 156025 from 780 to get -155245.

$$2x^{2}+790x=-155245$$

Divide both sides by 2.

$$\frac{2x^{2}+790x}{2}=-\frac{155245}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{790}{2}x=-\frac{155245}{2}$$

Divide 790 by 2.

$$x^{2}+395x=-\frac{155245}{2}$$

Divide 395, the coefficient of the x term, by 2 to get \frac{395}{2}. Then add the square of \frac{395}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+395x+\left(\frac{395}{2}\right)^{2}=-\frac{155245}{2}+\left(\frac{395}{2}\right)^{2}$$

Square \frac{395}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+395x+\frac{156025}{4}=-\frac{155245}{2}+\frac{156025}{4}$$

Add -\frac{155245}{2} to \frac{156025}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+395x+\frac{156025}{4}=-\frac{154465}{4}$$

Factor x^{2}+395x+\frac{156025}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{395}{2}\right)^{2}=-\frac{154465}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{395}{2}\right)^{2}}=\sqrt{-\frac{154465}{4}}$$

Simplify.

$$x+\frac{395}{2}=\frac{\sqrt{154465}i}{2} x+\frac{395}{2}=-\frac{\sqrt{154465}i}{2}$$

Subtract \frac{395}{2} from both sides of the equation.

$$x=\frac{-395+\sqrt{154465}i}{2} x=\frac{-\sqrt{154465}i-395}{2}$$