Solve 10^2+x^2=8^2-(12-x)^2

$$10 ^ { 2 } + x ^ { 2 } = 8 ^ { 2 } - ( 12 - x ) ^ { 2 }$$

Solve for x (complex solution)

$$x=6+3\sqrt{6}i\approx 6+7.348469228i$$

$$x=-3\sqrt{6}i+6\approx 6-7.348469228i$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 2 and get 100.

$$100+x^{2}=8^{2}-\left(12-x\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$100+x^{2}=64-\left(12-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-x\right)^{2}.

$$100+x^{2}=64-\left(144-24x+x^{2}\right)$$

To find the opposite of 144-24x+x^{2}, find the opposite of each term.

$$100+x^{2}=64-144+24x-x^{2}$$

Subtract 144 from 64 to get -80.

$$100+x^{2}=-80+24x-x^{2}$$

Subtract -80 from both sides.

$$100+x^{2}-\left(-80\right)=24x-x^{2}$$

The opposite of -80 is 80.

$$100+x^{2}+80=24x-x^{2}$$

Subtract 24x from both sides.

$$100+x^{2}+80-24x=-x^{2}$$

Add 100 and 80 to get 180.

$$180+x^{2}-24x=-x^{2}$$

Add x^{2} to both sides.

$$180+x^{2}-24x+x^{2}=0$$

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

$$180+2x^{2}-24x=0$$

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.

$$2x^{2}-24x+180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -24 for b, and 180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\times 180}}{2\times 2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 2\times 180}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-24\right)±\sqrt{576-8\times 180}}{2\times 2}$$

Multiply -8 times 180.

$$x=\frac{-\left(-24\right)±\sqrt{576-1440}}{2\times 2}$$

Add 576 to -1440.

$$x=\frac{-\left(-24\right)±\sqrt{-864}}{2\times 2}$$

Take the square root of -864.

$$x=\frac{-\left(-24\right)±12\sqrt{6}i}{2\times 2}$$

The opposite of -24 is 24.

$$x=\frac{24±12\sqrt{6}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{24±12\sqrt{6}i}{4}$$

Now solve the equation x=\frac{24±12\sqrt{6}i}{4} when ± is plus. Add 24 to 12i\sqrt{6}.

$$x=\frac{24+12\sqrt{6}i}{4}$$

Divide 24+12i\sqrt{6} by 4.

$$x=6+3\sqrt{6}i$$

Now solve the equation x=\frac{24±12\sqrt{6}i}{4} when ± is minus. Subtract 12i\sqrt{6} from 24.

$$x=\frac{-12\sqrt{6}i+24}{4}$$

Divide 24-12i\sqrt{6} by 4.

$$x=-3\sqrt{6}i+6$$

The equation is now solved.

$$x=6+3\sqrt{6}i x=-3\sqrt{6}i+6$$

Steps for Completing the Square

Calculate 10 to the power of 2 and get 100.

$$100+x^{2}=8^{2}-\left(12-x\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$100+x^{2}=64-\left(12-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-x\right)^{2}.

$$100+x^{2}=64-\left(144-24x+x^{2}\right)$$

To find the opposite of 144-24x+x^{2}, find the opposite of each term.

$$100+x^{2}=64-144+24x-x^{2}$$

Subtract 144 from 64 to get -80.

$$100+x^{2}=-80+24x-x^{2}$$

Subtract 24x from both sides.

$$100+x^{2}-24x=-80-x^{2}$$

Add x^{2} to both sides.

$$100+x^{2}-24x+x^{2}=-80$$

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

$$100+2x^{2}-24x=-80$$

Subtract 100 from both sides.

$$2x^{2}-24x=-80-100$$

Subtract 100 from -80 to get -180.

$$2x^{2}-24x=-180$$

Divide both sides by 2.

$$\frac{2x^{2}-24x}{2}=-\frac{180}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=-\frac{180}{2}$$

Divide -24 by 2.

$$x^{2}-12x=-\frac{180}{2}$$

Divide -180 by 2.

$$x^{2}-12x=-90$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-90+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-90+36$$

Add -90 to 36.

$$x^{2}-12x+36=-54$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=-54$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{-54}$$

Simplify.

$$x-6=3\sqrt{6}i x-6=-3\sqrt{6}i$$

Add 6 to both sides of the equation.

$$x=6+3\sqrt{6}i x=-3\sqrt{6}i+6$$