Solve (11+x)^2+(11-x)^2=120

$$( 11 + x ) ^ { 2 } + ( 11 - x ) ^ { 2 } = 120$$

Solve for x (complex solution)

$$x=-\sqrt{61}i\approx -0-7.810249676i$$

$$x=\sqrt{61}i\approx 7.810249676i$$

Steps by Finding Square Root

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(11+x\right)^{2}.

$$121+22x+x^{2}+\left(11-x\right)^{2}=120$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-x\right)^{2}.

$$121+22x+x^{2}+121-22x+x^{2}=120$$

Add 121 and 121 to get 242.

$$242+22x+x^{2}-22x+x^{2}=120$$

Combine 22x and -22x to get 0.

$$242+x^{2}+x^{2}=120$$

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

$$242+2x^{2}=120$$

Subtract 242 from both sides.

$$2x^{2}=120-242$$

Subtract 242 from 120 to get -122.

$$2x^{2}=-122$$

Divide both sides by 2.

$$x^{2}=\frac{-122}{2}$$

Divide -122 by 2 to get -61.

$$x^{2}=-61$$

The equation is now solved.

$$x=\sqrt{61}i x=-\sqrt{61}i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(11+x\right)^{2}.

$$121+22x+x^{2}+\left(11-x\right)^{2}=120$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-x\right)^{2}.

$$121+22x+x^{2}+121-22x+x^{2}=120$$

Add 121 and 121 to get 242.

$$242+22x+x^{2}-22x+x^{2}=120$$

Combine 22x and -22x to get 0.

$$242+x^{2}+x^{2}=120$$

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

$$242+2x^{2}=120$$

Subtract 120 from both sides.

$$242+2x^{2}-120=0$$

Subtract 120 from 242 to get 122.

$$122+2x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$2x^{2}+122=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and 122 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\times 122}}{2\times 2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 2\times 122}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{0±\sqrt{-8\times 122}}{2\times 2}$$

Multiply -8 times 122.

$$x=\frac{0±\sqrt{-976}}{2\times 2}$$

Take the square root of -976.

$$x=\frac{0±4\sqrt{61}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0±4\sqrt{61}i}{4}$$

Now solve the equation x=\frac{0±4\sqrt{61}i}{4} when ± is plus.

$$x=\sqrt{61}i$$

Now solve the equation x=\frac{0±4\sqrt{61}i}{4} when ± is minus.

$$x=-\sqrt{61}i$$

The equation is now solved.

$$x=\sqrt{61}i x=-\sqrt{61}i$$