Solve -112=x^2+x^2-32x

$$-112= { x }^{ 2 } + { x }^{ 2 } -32x$$

Solve for x

$$x=2\sqrt{2}+8\approx 10.828427125$$

$$x=8-2\sqrt{2}\approx 5.171572875$$

Steps Using the Quadratic Formula

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

$$-112=2x^{2}-32x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-32x=-112$$

Add 112 to both sides.

$$2x^{2}-32x+112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -32 for b, and 112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 2\times 112}}{2\times 2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 2\times 112}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-32\right)±\sqrt{1024-8\times 112}}{2\times 2}$$

Multiply -8 times 112.

$$x=\frac{-\left(-32\right)±\sqrt{1024-896}}{2\times 2}$$

Add 1024 to -896.

$$x=\frac{-\left(-32\right)±\sqrt{128}}{2\times 2}$$

Take the square root of 128.

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

The opposite of -32 is 32.

$$x=\frac{32±8\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{32±8\sqrt{2}}{4}$$

Now solve the equation x=\frac{32±8\sqrt{2}}{4} when ± is plus. Add 32 to 8\sqrt{2}.

$$x=\frac{8\sqrt{2}+32}{4}$$

Divide 32+8\sqrt{2} by 4.

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

Now solve the equation x=\frac{32±8\sqrt{2}}{4} when ± is minus. Subtract 8\sqrt{2} from 32.

$$x=\frac{32-8\sqrt{2}}{4}$$

Divide 32-8\sqrt{2} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$-112=2x^{2}-32x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-32x=-112$$

Divide both sides by 2.

$$\frac{2x^{2}-32x}{2}=-\frac{112}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{32}{2}\right)x=-\frac{112}{2}$$

Divide -32 by 2.

$$x^{2}-16x=-\frac{112}{2}$$

Divide -112 by 2.

$$x^{2}-16x=-56$$

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.

$$x^{2}-16x+\left(-8\right)^{2}=-56+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-56+64$$

Add -56 to 64.

$$x^{2}-16x+64=8$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 8 to both sides of the equation.

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