Solve -712=n(113+(n-1)-21)/2

$$- 712 = \frac { n ( 113 + ( n - 1 ) - 21 ) } { 2 }$$

Solve for n

$$n=\frac{\sqrt{2585}-91}{2}\approx -20.078552362$$

$$n=\frac{-\sqrt{2585}-91}{2}\approx -70.921447638$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$-712\times 2=n\left(113+n-1-21\right)$$

Multiply -712 and 2 to get -1424.

$$-1424=n\left(113+n-1-21\right)$$

Subtract 1 from 113 to get 112.

$$-1424=n\left(112+n-21\right)$$

Subtract 21 from 112 to get 91.

$$-1424=n\left(91+n\right)$$

Use the distributive property to multiply n by 91+n.

$$-1424=91n+n^{2}$$

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

$$91n+n^{2}=-1424$$

Add 1424 to both sides.

$$91n+n^{2}+1424=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$n^{2}+91n+1424=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 91 for b, and 1424 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-91±\sqrt{91^{2}-4\times 1424}}{2}$$

Square 91.

$$n=\frac{-91±\sqrt{8281-4\times 1424}}{2}$$

Multiply -4 times 1424.

$$n=\frac{-91±\sqrt{8281-5696}}{2}$$

Add 8281 to -5696.

$$n=\frac{-91±\sqrt{2585}}{2}$$

Now solve the equation n=\frac{-91±\sqrt{2585}}{2} when ± is plus. Add -91 to \sqrt{2585}.

$$n=\frac{\sqrt{2585}-91}{2}$$

Now solve the equation n=\frac{-91±\sqrt{2585}}{2} when ± is minus. Subtract \sqrt{2585} from -91.

$$n=\frac{-\sqrt{2585}-91}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{2585}-91}{2} n=\frac{-\sqrt{2585}-91}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$-712\times 2=n\left(113+n-1-21\right)$$

Multiply -712 and 2 to get -1424.

$$-1424=n\left(113+n-1-21\right)$$

Subtract 1 from 113 to get 112.

$$-1424=n\left(112+n-21\right)$$

Subtract 21 from 112 to get 91.

$$-1424=n\left(91+n\right)$$

Use the distributive property to multiply n by 91+n.

$$-1424=91n+n^{2}$$

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

$$91n+n^{2}=-1424$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$n^{2}+91n=-1424$$

Divide 91, the coefficient of the x term, by 2 to get \frac{91}{2}. Then add the square of \frac{91}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+91n+\left(\frac{91}{2}\right)^{2}=-1424+\left(\frac{91}{2}\right)^{2}$$

Square \frac{91}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+91n+\frac{8281}{4}=-1424+\frac{8281}{4}$$

Add -1424 to \frac{8281}{4}.

$$n^{2}+91n+\frac{8281}{4}=\frac{2585}{4}$$

Factor n^{2}+91n+\frac{8281}{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}.

$$\left(n+\frac{91}{2}\right)^{2}=\frac{2585}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{91}{2}\right)^{2}}=\sqrt{\frac{2585}{4}}$$

Simplify.

$$n+\frac{91}{2}=\frac{\sqrt{2585}}{2} n+\frac{91}{2}=-\frac{\sqrt{2585}}{2}$$

Subtract \frac{91}{2} from both sides of the equation.

$$n=\frac{\sqrt{2585}-91}{2} n=\frac{-\sqrt{2585}-91}{2}$$