Solve (n+3)(n-5)=20

$$( n + 3 ) ( n - 5 ) = 20$$

Solve for n

$$n=-5$$

$$n=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n+3 by n-5 and combine like terms.

$$n^{2}-2n-15=20$$

Subtract 20 from both sides.

$$n^{2}-2n-15-20=0$$

Subtract 20 from -15 to get -35.

$$n^{2}-2n-35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-35\right)}}{2}$$

Square -2.

$$n=\frac{-\left(-2\right)±\sqrt{4-4\left(-35\right)}}{2}$$

Multiply -4 times -35.

$$n=\frac{-\left(-2\right)±\sqrt{4+140}}{2}$$

Add 4 to 140.

$$n=\frac{-\left(-2\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$n=\frac{-\left(-2\right)±12}{2}$$

The opposite of -2 is 2.

$$n=\frac{2±12}{2}$$

Now solve the equation n=\frac{2±12}{2} when ± is plus. Add 2 to 12.

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

Divide 14 by 2.

$$n=7$$

Now solve the equation n=\frac{2±12}{2} when ± is minus. Subtract 12 from 2.

$$n=-\frac{10}{2}$$

Divide -10 by 2.

$$n=-5$$

The equation is now solved.

$$n=7 n=-5$$

Steps for Completing the Square

Use the distributive property to multiply n+3 by n-5 and combine like terms.

$$n^{2}-2n-15=20$$

Add 15 to both sides.

$$n^{2}-2n=20+15$$

Add 20 and 15 to get 35.

$$n^{2}-2n=35$$

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

$$n^{2}-2n+1=35+1$$

Add 35 to 1.

$$n^{2}-2n+1=36$$

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

$$\left(n-1\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-1\right)^{2}}=\sqrt{36}$$

Simplify.

$$n-1=6 n-1=-6$$

Add 1 to both sides of the equation.

$$n=7 n=-5$$