Solve (n+3)(n+2)=90

$$( n + 3 ) ( n + 2 ) = 90$$

Solve for n

$$n=-12$$

$$n=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n+3 by n+2 and combine like terms.

$$n^{2}+5n+6=90$$

Subtract 90 from both sides.

$$n^{2}+5n+6-90=0$$

Subtract 90 from 6 to get -84.

$$n^{2}+5n-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-5±\sqrt{5^{2}-4\left(-84\right)}}{2}$$

Square 5.

$$n=\frac{-5±\sqrt{25-4\left(-84\right)}}{2}$$

Multiply -4 times -84.

$$n=\frac{-5±\sqrt{25+336}}{2}$$

Add 25 to 336.

$$n=\frac{-5±\sqrt{361}}{2}$$

Take the square root of 361.

$$n=\frac{-5±19}{2}$$

Now solve the equation n=\frac{-5±19}{2} when ± is plus. Add -5 to 19.

$$n=\frac{14}{2}$$

Divide 14 by 2.

$$n=7$$

Now solve the equation n=\frac{-5±19}{2} when ± is minus. Subtract 19 from -5.

$$n=-\frac{24}{2}$$

Divide -24 by 2.

$$n=-12$$

The equation is now solved.

$$n=7 n=-12$$

Steps for Completing the Square

Use the distributive property to multiply n+3 by n+2 and combine like terms.

$$n^{2}+5n+6=90$$

Subtract 6 from both sides.

$$n^{2}+5n=90-6$$

Subtract 6 from 90 to get 84.

$$n^{2}+5n=84$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+5n+\left(\frac{5}{2}\right)^{2}=84+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+5n+\frac{25}{4}=84+\frac{25}{4}$$

Add 84 to \frac{25}{4}.

$$n^{2}+5n+\frac{25}{4}=\frac{361}{4}$$

Factor n^{2}+5n+\frac{25}{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}.

$$\left(n+\frac{5}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{5}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$n+\frac{5}{2}=\frac{19}{2} n+\frac{5}{2}=-\frac{19}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$n=7 n=-12$$