Solve x^2-16x+64=32

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

Solve for x

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

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

Steps Using the Quadratic Formula

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.

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

Subtract 32 from both sides of the equation.

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

Subtracting 32 from itself leaves 0.

$$x^{2}-16x+64-32=0$$

Subtract 32 from 64.

$$x^{2}-16x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 32}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 32}}{2}$$

Multiply -4 times 32.

$$x=\frac{-\left(-16\right)±\sqrt{256-128}}{2}$$

Add 256 to -128.

$$x=\frac{-\left(-16\right)±\sqrt{128}}{2}$$

Take the square root of 128.

$$x=\frac{-\left(-16\right)±8\sqrt{2}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±8\sqrt{2}}{2}$$

Now solve the equation x=\frac{16±8\sqrt{2}}{2} when ± is plus. Add 16 to 8\sqrt{2}.

$$x=\frac{8\sqrt{2}+16}{2}$$

Divide 16+8\sqrt{2} by 2.

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

Now solve the equation x=\frac{16±8\sqrt{2}}{2} when ± is minus. Subtract 8\sqrt{2} from 16.

$$x=\frac{16-8\sqrt{2}}{2}$$

Divide 16-8\sqrt{2} by 2.

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

The equation is now solved.

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

Steps for Completing the Square

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.

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

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.

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