Solve (x-2)(x+1)=16

$$(x-2)(x+1)=16$$

Solve for x

$$x = \frac{\sqrt{73} + 1}{2} \approx 4.772001873$$

$$x=\frac{1-\sqrt{73}}{2}\approx -3.772001873$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2 by x+1 and combine like terms.

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

Subtract 16 from both sides.

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

Subtract 16 from -2 to get -18.

$$x^{2}-x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-\left(-1\right)±\sqrt{1+72}}{2}$$

Add 1 to 72.

$$x=\frac{-\left(-1\right)±\sqrt{73}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{73}}{2}$$

Now solve the equation x=\frac{1±\sqrt{73}}{2} when ± is plus. Add 1 to \sqrt{73}.

$$x=\frac{\sqrt{73}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{73}}{2} when ± is minus. Subtract \sqrt{73} from 1.

$$x=\frac{1-\sqrt{73}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{73}+1}{2} x=\frac{1-\sqrt{73}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-2 by x+1 and combine like terms.

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

Add 2 to both sides.

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

Add 16 and 2 to get 18.

$$x^{2}-x=18$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=18+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=18+\frac{1}{4}$$

Add 18 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{73}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{73}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{73}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{73}}{2} x-\frac{1}{2}=-\frac{\sqrt{73}}{2}$$

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

$$x=\frac{\sqrt{73}+1}{2} x=\frac{1-\sqrt{73}}{2}$$