Solve x^2+x^2=(6-x)^2

$$x ^ { 2 } + x ^ { 2 } = ( 6 - x ) ^ { 2 }$$

Solve for x

$$x=6\sqrt{2}-6\approx 2.485281374$$

$$x=-6\sqrt{2}-6\approx -14.485281374$$

Steps Using the Quadratic Formula

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

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

Subtract 36 from both sides.

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

Add 12x to both sides.

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

Subtract x^{2} from both sides.

$$2x^{2}-36+12x-x^{2}=0$$

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

$$x^{2}-36+12x=0$$

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.

$$x^{2}+12x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-36\right)}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$x=\frac{-12±\sqrt{144+144}}{2}$$

Add 144 to 144.

$$x=\frac{-12±\sqrt{288}}{2}$$

Take the square root of 288.

$$x=\frac{-12±12\sqrt{2}}{2}$$

Now solve the equation x=\frac{-12±12\sqrt{2}}{2} when ± is plus. Add -12 to 12\sqrt{2}.

$$x=\frac{12\sqrt{2}-12}{2}$$

Divide -12+12\sqrt{2} by 2.

$$x=6\sqrt{2}-6$$

Now solve the equation x=\frac{-12±12\sqrt{2}}{2} when ± is minus. Subtract 12\sqrt{2} from -12.

$$x=\frac{-12\sqrt{2}-12}{2}$$

Divide -12-12\sqrt{2} by 2.

$$x=-6\sqrt{2}-6$$

The equation is now solved.

$$x=6\sqrt{2}-6 x=-6\sqrt{2}-6$$

Steps for Completing the Square

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

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

Add 12x to both sides.

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

Subtract x^{2} from both sides.

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

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

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

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.

$$x^{2}+12x+6^{2}=36+6^{2}$$

Square 6.

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

Add 36 to 36.

$$x^{2}+12x+36=72$$

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}.

$$\left(x+6\right)^{2}=72$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{72}$$

Simplify.

$$x+6=6\sqrt{2} x+6=-6\sqrt{2}$$

Subtract 6 from both sides of the equation.

$$x=6\sqrt{2}-6 x=-6\sqrt{2}-6$$