Solve (n+2)(n+1)=145

$$( n + 2 ) ( n + 1 ) = 145$$

Solve for n

$$n = \frac{\sqrt{581} - 3}{2} \approx 10.551970793$$

$$n=\frac{-\sqrt{581}-3}{2}\approx -13.551970793$$

Steps Using the Quadratic Formula

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

$$n^{2}+3n+2=145$$

Subtract 145 from both sides.

$$n^{2}+3n+2-145=0$$

Subtract 145 from 2 to get -143.

$$n^{2}+3n-143=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -143 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-3±\sqrt{3^{2}-4\left(-143\right)}}{2}$$

Square 3.

$$n=\frac{-3±\sqrt{9-4\left(-143\right)}}{2}$$

Multiply -4 times -143.

$$n=\frac{-3±\sqrt{9+572}}{2}$$

Add 9 to 572.

$$n=\frac{-3±\sqrt{581}}{2}$$

Now solve the equation n=\frac{-3±\sqrt{581}}{2} when ± is plus. Add -3 to \sqrt{581}.

$$n=\frac{\sqrt{581}-3}{2}$$

Now solve the equation n=\frac{-3±\sqrt{581}}{2} when ± is minus. Subtract \sqrt{581} from -3.

$$n=\frac{-\sqrt{581}-3}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{581}-3}{2} n=\frac{-\sqrt{581}-3}{2}$$

Steps for Completing the Square

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

$$n^{2}+3n+2=145$$

Subtract 2 from both sides.

$$n^{2}+3n=145-2$$

Subtract 2 from 145 to get 143.

$$n^{2}+3n=143$$

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.

$$n^{2}+3n+\left(\frac{3}{2}\right)^{2}=143+\left(\frac{3}{2}\right)^{2}$$

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

$$n^{2}+3n+\frac{9}{4}=143+\frac{9}{4}$$

Add 143 to \frac{9}{4}.

$$n^{2}+3n+\frac{9}{4}=\frac{581}{4}$$

Factor n^{2}+3n+\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}.

$$\left(n+\frac{3}{2}\right)^{2}=\frac{581}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{3}{2}\right)^{2}}=\sqrt{\frac{581}{4}}$$

Simplify.

$$n+\frac{3}{2}=\frac{\sqrt{581}}{2} n+\frac{3}{2}=-\frac{\sqrt{581}}{2}$$

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

$$n=\frac{\sqrt{581}-3}{2} n=\frac{-\sqrt{581}-3}{2}$$