Solve -712=n(226+(n-1)-21)/2

$$- 712 = \frac { n ( 226 + ( n - 1 ) - 21 ) } { 2 }$$

Solve for n

$$n=2\sqrt{2245}-102\approx -7.237138076$$

$$n=-2\sqrt{2245}-102\approx -196.762861924$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$-712\times 2=n\left(226+n-1-21\right)$$

Multiply -712 and 2 to get -1424.

$$-1424=n\left(226+n-1-21\right)$$

Subtract 1 from 226 to get 225.

$$-1424=n\left(225+n-21\right)$$

Subtract 21 from 225 to get 204.

$$-1424=n\left(204+n\right)$$

Use the distributive property to multiply n by 204+n.

$$-1424=204n+n^{2}$$

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

$$204n+n^{2}=-1424$$

Add 1424 to both sides.

$$204n+n^{2}+1424=0$$

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.

$$n^{2}+204n+1424=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 204 for b, and 1424 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-204±\sqrt{204^{2}-4\times 1424}}{2}$$

Square 204.

$$n=\frac{-204±\sqrt{41616-4\times 1424}}{2}$$

Multiply -4 times 1424.

$$n=\frac{-204±\sqrt{41616-5696}}{2}$$

Add 41616 to -5696.

$$n=\frac{-204±\sqrt{35920}}{2}$$

Take the square root of 35920.

$$n=\frac{-204±4\sqrt{2245}}{2}$$

Now solve the equation n=\frac{-204±4\sqrt{2245}}{2} when ± is plus. Add -204 to 4\sqrt{2245}.

$$n=\frac{4\sqrt{2245}-204}{2}$$

Divide -204+4\sqrt{2245} by 2.

$$n=2\sqrt{2245}-102$$

Now solve the equation n=\frac{-204±4\sqrt{2245}}{2} when ± is minus. Subtract 4\sqrt{2245} from -204.

$$n=\frac{-4\sqrt{2245}-204}{2}$$

Divide -204-4\sqrt{2245} by 2.

$$n=-2\sqrt{2245}-102$$

The equation is now solved.

$$n=2\sqrt{2245}-102 n=-2\sqrt{2245}-102$$

Steps for Completing the Square

Multiply both sides by 2.

$$-712\times 2=n\left(226+n-1-21\right)$$

Multiply -712 and 2 to get -1424.

$$-1424=n\left(226+n-1-21\right)$$

Subtract 1 from 226 to get 225.

$$-1424=n\left(225+n-21\right)$$

Subtract 21 from 225 to get 204.

$$-1424=n\left(204+n\right)$$

Use the distributive property to multiply n by 204+n.

$$-1424=204n+n^{2}$$

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

$$204n+n^{2}=-1424$$

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.

$$n^{2}+204n=-1424$$

Divide 204, the coefficient of the x term, by 2 to get 102. Then add the square of 102 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+204n+102^{2}=-1424+102^{2}$$

Square 102.

$$n^{2}+204n+10404=-1424+10404$$

Add -1424 to 10404.

$$n^{2}+204n+10404=8980$$

Factor n^{2}+204n+10404. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+102\right)^{2}=8980$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+102\right)^{2}}=\sqrt{8980}$$

Simplify.

$$n+102=2\sqrt{2245} n+102=-2\sqrt{2245}$$

Subtract 102 from both sides of the equation.

$$n=2\sqrt{2245}-102 n=-2\sqrt{2245}-102$$