Solve (x-5)(x+4)=112

$$(x-5)(x+4)=112$$

Solve for x

$$x=-11$$

$$x=12$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-5 by x+4 and combine like terms.

$$x^{2}-x-20=112$$

Subtract 112 from both sides.

$$x^{2}-x-20-112=0$$

Subtract 112 from -20 to get -132.

$$x^{2}-x-132=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -132 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-132\right)}}{2}$$

Multiply -4 times -132.

$$x=\frac{-\left(-1\right)±\sqrt{1+528}}{2}$$

Add 1 to 528.

$$x=\frac{-\left(-1\right)±\sqrt{529}}{2}$$

Take the square root of 529.

$$x=\frac{-\left(-1\right)±23}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±23}{2}$$

Now solve the equation x=\frac{1±23}{2} when ± is plus. Add 1 to 23.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{1±23}{2} when ± is minus. Subtract 23 from 1.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=12 x=-11$$

Steps for Completing the Square

Use the distributive property to multiply x-5 by x+4 and combine like terms.

$$x^{2}-x-20=112$$

Add 20 to both sides.

$$x^{2}-x=112+20$$

Add 112 and 20 to get 132.

$$x^{2}-x=132$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=132+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=132+\frac{1}{4}$$

Add 132 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{529}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{529}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{23}{2} x-\frac{1}{2}=-\frac{23}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=12 x=-11$$