Solve (n+1)n/2=2016

$$\frac { ( n + 1 ) n } { 2 } = 2016$$

Solve for n

$$n=63$$

$$n=-64$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$\left(n+1\right)n=2016\times 2$$

Use the distributive property to multiply n+1 by n.

$$n^{2}+n=2016\times 2$$

Multiply 2016 and 2 to get 4032.

$$n^{2}+n=4032$$

Subtract 4032 from both sides.

$$n^{2}+n-4032=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -4032 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-4032\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-4032\right)}}{2}$$

Multiply -4 times -4032.

$$n=\frac{-1±\sqrt{1+16128}}{2}$$

Add 1 to 16128.

$$n=\frac{-1±\sqrt{16129}}{2}$$

Take the square root of 16129.

$$n=\frac{-1±127}{2}$$

Now solve the equation n=\frac{-1±127}{2} when ± is plus. Add -1 to 127.

$$n=\frac{126}{2}$$

Divide 126 by 2.

$$n=63$$

Now solve the equation n=\frac{-1±127}{2} when ± is minus. Subtract 127 from -1.

$$n=-\frac{128}{2}$$

Divide -128 by 2.

$$n=-64$$

The equation is now solved.

$$n=63 n=-64$$

Steps for Completing the Square

Multiply both sides by 2.

$$\left(n+1\right)n=2016\times 2$$

Use the distributive property to multiply n+1 by n.

$$n^{2}+n=2016\times 2$$

Multiply 2016 and 2 to get 4032.

$$n^{2}+n=4032$$

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

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=4032+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+n+\frac{1}{4}=4032+\frac{1}{4}$$

Add 4032 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{16129}{4}$$

Factor n^{2}+n+\frac{1}{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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{16129}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{16129}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{127}{2} n+\frac{1}{2}=-\frac{127}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=63 n=-64$$