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

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

Solve for x

$$x=6\sqrt{2}+12\approx 20.485281374$$

$$x=12-6\sqrt{2}\approx 3.514718626$$

Steps Using the Quadratic Formula

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

$$2\left(36-12x+x^{2}\right)=x^{2}$$

Use the distributive property to multiply 2 by 36-12x+x^{2}.

$$72-24x+2x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$72-24x+2x^{2}-x^{2}=0$$

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

$$72-24x+x^{2}=0$$

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.

$$x^{2}-24x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 72}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 72}}{2}$$

Multiply -4 times 72.

$$x=\frac{-\left(-24\right)±\sqrt{576-288}}{2}$$

Add 576 to -288.

$$x=\frac{-\left(-24\right)±\sqrt{288}}{2}$$

Take the square root of 288.

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

The opposite of -24 is 24.

$$x=\frac{24±12\sqrt{2}}{2}$$

Now solve the equation x=\frac{24±12\sqrt{2}}{2} when ± is plus. Add 24 to 12\sqrt{2}.

$$x=\frac{12\sqrt{2}+24}{2}$$

Divide 24+12\sqrt{2} by 2.

$$x=6\sqrt{2}+12$$

Now solve the equation x=\frac{24±12\sqrt{2}}{2} when ± is minus. Subtract 12\sqrt{2} from 24.

$$x=\frac{24-12\sqrt{2}}{2}$$

Divide 24-12\sqrt{2} by 2.

$$x=12-6\sqrt{2}$$

The equation is now solved.

$$x=6\sqrt{2}+12 x=12-6\sqrt{2}$$

Steps for Completing the Square

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

$$2\left(36-12x+x^{2}\right)=x^{2}$$

Use the distributive property to multiply 2 by 36-12x+x^{2}.

$$72-24x+2x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$72-24x+2x^{2}-x^{2}=0$$

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

$$72-24x+x^{2}=0$$

Subtract 72 from both sides. Anything subtracted from zero gives its negation.

$$-24x+x^{2}=-72$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-24x=-72$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=-72+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-72+144$$

Add -72 to 144.

$$x^{2}-24x+144=72$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=72$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{72}$$

Simplify.

$$x-12=6\sqrt{2} x-12=-6\sqrt{2}$$

Add 12 to both sides of the equation.

$$x=6\sqrt{2}+12 x=12-6\sqrt{2}$$