Solve 2left(8-xright)^2=8^2

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

Solve for x

$$x=8-4\sqrt{2}\approx 2.343145751$$

$$x=4\sqrt{2}+8\approx 13.656854249$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-x\right)^{2}.

$$2\left(64-16x+x^{2}\right)=8^{2}$$

Use the distributive property to multiply 2 by 64-16x+x^{2}.

$$128-32x+2x^{2}=8^{2}$$

Calculate 8 to the power of 2 and get 64.

$$128-32x+2x^{2}=64$$

Subtract 64 from both sides.

$$128-32x+2x^{2}-64=0$$

Subtract 64 from 128 to get 64.

$$64-32x+2x^{2}=0$$

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.

$$2x^{2}-32x+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -32 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 2\times 64}}{2\times 2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 2\times 64}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-32\right)±\sqrt{1024-8\times 64}}{2\times 2}$$

Multiply -8 times 64.

$$x=\frac{-\left(-32\right)±\sqrt{1024-512}}{2\times 2}$$

Add 1024 to -512.

$$x=\frac{-\left(-32\right)±\sqrt{512}}{2\times 2}$$

Take the square root of 512.

$$x=\frac{-\left(-32\right)±16\sqrt{2}}{2\times 2}$$

The opposite of -32 is 32.

$$x=\frac{32±16\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{32±16\sqrt{2}}{4}$$

Now solve the equation x=\frac{32±16\sqrt{2}}{4} when ± is plus. Add 32 to 16\sqrt{2}.

$$x=\frac{16\sqrt{2}+32}{4}$$

Divide 32+16\sqrt{2} by 4.

$$x=4\sqrt{2}+8$$

Now solve the equation x=\frac{32±16\sqrt{2}}{4} when ± is minus. Subtract 16\sqrt{2} from 32.

$$x=\frac{32-16\sqrt{2}}{4}$$

Divide 32-16\sqrt{2} by 4.

$$x=8-4\sqrt{2}$$

The equation is now solved.

$$x=4\sqrt{2}+8 x=8-4\sqrt{2}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-x\right)^{2}.

$$2\left(64-16x+x^{2}\right)=8^{2}$$

Use the distributive property to multiply 2 by 64-16x+x^{2}.

$$128-32x+2x^{2}=8^{2}$$

Calculate 8 to the power of 2 and get 64.

$$128-32x+2x^{2}=64$$

Subtract 128 from both sides.

$$-32x+2x^{2}=64-128$$

Subtract 128 from 64 to get -64.

$$-32x+2x^{2}=-64$$

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.

$$2x^{2}-32x=-64$$

Divide both sides by 2.

$$\frac{2x^{2}-32x}{2}=-\frac{64}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{32}{2}\right)x=-\frac{64}{2}$$

Divide -32 by 2.

$$x^{2}-16x=-\frac{64}{2}$$

Divide -64 by 2.

$$x^{2}-16x=-32$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-32+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-32+64$$

Add -32 to 64.

$$x^{2}-16x+64=32$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=32$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{32}$$

Simplify.

$$x-8=4\sqrt{2} x-8=-4\sqrt{2}$$

Add 8 to both sides of the equation.

$$x=4\sqrt{2}+8 x=8-4\sqrt{2}$$