Solve 220=1/2n(n+2)

$$220 = \frac { 1 } { 2 } n ( n + 2 )$$

Solve for n

$$n=-22$$

$$n=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2}n by n+2.

$$220=\frac{1}{2}nn+\frac{1}{2}n\times 2$$

Multiply n and n to get n^{2}.

$$220=\frac{1}{2}n^{2}+\frac{1}{2}n\times 2$$

Cancel out 2 and 2.

$$220=\frac{1}{2}n^{2}+n$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}n^{2}+n=220$$

Subtract 220 from both sides.

$$\frac{1}{2}n^{2}+n-220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 1 for b, and -220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{2}\left(-220\right)}}{2\times \frac{1}{2}}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\times \frac{1}{2}\left(-220\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$n=\frac{-1±\sqrt{1-2\left(-220\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -220.

$$n=\frac{-1±\sqrt{1+440}}{2\times \frac{1}{2}}$$

Add 1 to 440.

$$n=\frac{-1±\sqrt{441}}{2\times \frac{1}{2}}$$

Take the square root of 441.

$$n=\frac{-1±21}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$n=\frac{-1±21}{1}$$

Now solve the equation n=\frac{-1±21}{1} when ± is plus. Add -1 to 21.

$$n=\frac{20}{1}$$

Divide 20 by 1.

$$n=20$$

Now solve the equation n=\frac{-1±21}{1} when ± is minus. Subtract 21 from -1.

$$n=-\frac{22}{1}$$

Divide -22 by 1.

$$n=-22$$

The equation is now solved.

$$n=20 n=-22$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2}n by n+2.

$$220=\frac{1}{2}nn+\frac{1}{2}n\times 2$$

Multiply n and n to get n^{2}.

$$220=\frac{1}{2}n^{2}+\frac{1}{2}n\times 2$$

Cancel out 2 and 2.

$$220=\frac{1}{2}n^{2}+n$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}n^{2}+n=220$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}n^{2}+n}{\frac{1}{2}}=\frac{220}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$n^{2}+\frac{1}{\frac{1}{2}}n=\frac{220}{\frac{1}{2}}$$

Divide 1 by \frac{1}{2} by multiplying 1 by the reciprocal of \frac{1}{2}.

$$n^{2}+2n=\frac{220}{\frac{1}{2}}$$

Divide 220 by \frac{1}{2} by multiplying 220 by the reciprocal of \frac{1}{2}.

$$n^{2}+2n=440$$

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.

$$n^{2}+2n+1^{2}=440+1^{2}$$

Square 1.

$$n^{2}+2n+1=440+1$$

Add 440 to 1.

$$n^{2}+2n+1=441$$

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}.

$$\left(n+1\right)^{2}=441$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+1\right)^{2}}=\sqrt{441}$$

Simplify.

$$n+1=21 n+1=-21$$

Subtract 1 from both sides of the equation.

$$n=20 n=-22$$