Solve x^2-20x=32

$$x ^ { 2 } - 20 x = 32$$

Solve for x

$$x=2\sqrt{33}+10\approx 21.489125293$$

$$x=10-2\sqrt{33}\approx -1.489125293$$

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.

$$x^{2}-20x=32$$

Subtract 32 from both sides of the equation.

$$x^{2}-20x-32=32-32$$

Subtracting 32 from itself leaves 0.

$$x^{2}-20x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-32\right)}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$x=\frac{-\left(-20\right)±\sqrt{400+128}}{2}$$

Add 400 to 128.

$$x=\frac{-\left(-20\right)±\sqrt{528}}{2}$$

Take the square root of 528.

$$x=\frac{-\left(-20\right)±4\sqrt{33}}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±4\sqrt{33}}{2}$$

Now solve the equation x=\frac{20±4\sqrt{33}}{2} when ± is plus. Add 20 to 4\sqrt{33}.

$$x=\frac{4\sqrt{33}+20}{2}$$

Divide 20+4\sqrt{33} by 2.

$$x=2\sqrt{33}+10$$

Now solve the equation x=\frac{20±4\sqrt{33}}{2} when ± is minus. Subtract 4\sqrt{33} from 20.

$$x=\frac{20-4\sqrt{33}}{2}$$

Divide 20-4\sqrt{33} by 2.

$$x=10-2\sqrt{33}$$

The equation is now solved.

$$x=2\sqrt{33}+10 x=10-2\sqrt{33}$$

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.

$$x^{2}-20x=32$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=32+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=32+100$$

Add 32 to 100.

$$x^{2}-20x+100=132$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=132$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{132}$$

Simplify.

$$x-10=2\sqrt{33} x-10=-2\sqrt{33}$$

Add 10 to both sides of the equation.

$$x=2\sqrt{33}+10 x=10-2\sqrt{33}$$