Solve (x+1)(x+2)=9

$$( x + 1 ) ( x + 2 ) = 9$$

Solve for x

$$x = \frac{\sqrt{37} - 3}{2} \approx 1.541381265$$

$$x=\frac{-\sqrt{37}-3}{2}\approx -4.541381265$$

Steps Using the Quadratic Formula

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

$$x^{2}+3x+2=9$$

Subtract 9 from both sides.

$$x^{2}+3x+2-9=0$$

Subtract 9 from 2 to get -7.

$$x^{2}+3x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-7\right)}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$x=\frac{-3±\sqrt{9+28}}{2}$$

Add 9 to 28.

$$x=\frac{-3±\sqrt{37}}{2}$$

Now solve the equation x=\frac{-3±\sqrt{37}}{2} when ± is plus. Add -3 to \sqrt{37}.

$$x=\frac{\sqrt{37}-3}{2}$$

Now solve the equation x=\frac{-3±\sqrt{37}}{2} when ± is minus. Subtract \sqrt{37} from -3.

$$x=\frac{-\sqrt{37}-3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{37}-3}{2} x=\frac{-\sqrt{37}-3}{2}$$

Steps for Completing the Square

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

$$x^{2}+3x+2=9$$

Subtract 2 from both sides.

$$x^{2}+3x=9-2$$

Subtract 2 from 9 to get 7.

$$x^{2}+3x=7$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=7+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=7+\frac{9}{4}$$

Add 7 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{37}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{37}}{2} x+\frac{3}{2}=-\frac{\sqrt{37}}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{37}-3}{2} x=\frac{-\sqrt{37}-3}{2}$$