Solve (x+4)^2+(x-5)^2=2^2

$$( x + 4 ) ^ { 2 } + ( x - 5 ) ^ { 2 } = 2 ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{1+\sqrt{73}i}{2}\approx 0.5+4.272001873i$$

$$x=\frac{-\sqrt{73}i+1}{2}\approx 0.5-4.272001873i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{2}+8x+16+\left(x-5\right)^{2}=2^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}+8x+16+x^{2}-10x+25=2^{2}$$

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

$$2x^{2}+8x+16-10x+25=2^{2}$$

Combine 8x and -10x to get -2x.

$$2x^{2}-2x+16+25=2^{2}$$

Add 16 and 25 to get 41.

$$2x^{2}-2x+41=2^{2}$$

Calculate 2 to the power of 2 and get 4.

$$2x^{2}-2x+41=4$$

Subtract 4 from both sides.

$$2x^{2}-2x+41-4=0$$

Subtract 4 from 41 to get 37.

$$2x^{2}-2x+37=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and 37 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\times 37}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\times 37}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\times 37}}{2\times 2}$$

Multiply -8 times 37.

$$x=\frac{-\left(-2\right)±\sqrt{4-296}}{2\times 2}$$

Add 4 to -296.

$$x=\frac{-\left(-2\right)±\sqrt{-292}}{2\times 2}$$

Take the square root of -292.

$$x=\frac{-\left(-2\right)±2\sqrt{73}i}{2\times 2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{73}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±2\sqrt{73}i}{4}$$

Now solve the equation x=\frac{2±2\sqrt{73}i}{4} when ± is plus. Add 2 to 2i\sqrt{73}.

$$x=\frac{2+2\sqrt{73}i}{4}$$

Divide 2+2i\sqrt{73} by 4.

$$x=\frac{1+\sqrt{73}i}{2}$$

Now solve the equation x=\frac{2±2\sqrt{73}i}{4} when ± is minus. Subtract 2i\sqrt{73} from 2.

$$x=\frac{-2\sqrt{73}i+2}{4}$$

Divide 2-2i\sqrt{73} by 4.

$$x=\frac{-\sqrt{73}i+1}{2}$$

The equation is now solved.

$$x=\frac{1+\sqrt{73}i}{2} x=\frac{-\sqrt{73}i+1}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{2}+8x+16+\left(x-5\right)^{2}=2^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}+8x+16+x^{2}-10x+25=2^{2}$$

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

$$2x^{2}+8x+16-10x+25=2^{2}$$

Combine 8x and -10x to get -2x.

$$2x^{2}-2x+16+25=2^{2}$$

Add 16 and 25 to get 41.

$$2x^{2}-2x+41=2^{2}$$

Calculate 2 to the power of 2 and get 4.

$$2x^{2}-2x+41=4$$

Subtract 41 from both sides.

$$2x^{2}-2x=4-41$$

Subtract 41 from 4 to get -37.

$$2x^{2}-2x=-37$$

Divide both sides by 2.

$$\frac{2x^{2}-2x}{2}=-\frac{37}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{2}{2}\right)x=-\frac{37}{2}$$

Divide -2 by 2.

$$x^{2}-x=-\frac{37}{2}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{37}{2}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=-\frac{37}{2}+\frac{1}{4}$$

Add -\frac{37}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=-\frac{73}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{73}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{73}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{73}i}{2} x-\frac{1}{2}=-\frac{\sqrt{73}i}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{1+\sqrt{73}i}{2} x=\frac{-\sqrt{73}i+1}{2}$$