Solve -30=5n+n(n-1)/2(-1)

$$- 30 = 5 n + \frac { n ( n - 1 ) } { 2 } ( - 1 )$$

Solve for n

$$n=-4$$

$$n=15$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$-60=10n+n\left(n-1\right)\left(-1\right)$$

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

$$-60=10n+\left(n^{2}-n\right)\left(-1\right)$$

Use the distributive property to multiply n^{2}-n by -1.

$$-60=10n-n^{2}+n$$

Combine 10n and n to get 11n.

$$-60=11n-n^{2}$$

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

$$11n-n^{2}=-60$$

Add 60 to both sides.

$$11n-n^{2}+60=0$$

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.

$$-n^{2}+11n+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\times 60}}{2\left(-1\right)}$$

Square 11.

$$n=\frac{-11±\sqrt{121-4\left(-1\right)\times 60}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-11±\sqrt{121+4\times 60}}{2\left(-1\right)}$$

Multiply 4 times 60.

$$n=\frac{-11±\sqrt{121+240}}{2\left(-1\right)}$$

Add 121 to 240.

$$n=\frac{-11±\sqrt{361}}{2\left(-1\right)}$$

Take the square root of 361.

$$n=\frac{-11±19}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-11±19}{-2}$$

Now solve the equation n=\frac{-11±19}{-2} when ± is plus. Add -11 to 19.

$$n=\frac{8}{-2}$$

Divide 8 by -2.

$$n=-4$$

Now solve the equation n=\frac{-11±19}{-2} when ± is minus. Subtract 19 from -11.

$$n=-\frac{30}{-2}$$

Divide -30 by -2.

$$n=15$$

The equation is now solved.

$$n=-4 n=15$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$-60=10n+n\left(n-1\right)\left(-1\right)$$

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

$$-60=10n+\left(n^{2}-n\right)\left(-1\right)$$

Use the distributive property to multiply n^{2}-n by -1.

$$-60=10n-n^{2}+n$$

Combine 10n and n to get 11n.

$$-60=11n-n^{2}$$

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

$$11n-n^{2}=-60$$

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.

$$-n^{2}+11n=-60$$

Divide both sides by -1.

$$\frac{-n^{2}+11n}{-1}=-\frac{60}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{11}{-1}n=-\frac{60}{-1}$$

Divide 11 by -1.

$$n^{2}-11n=-\frac{60}{-1}$$

Divide -60 by -1.

$$n^{2}-11n=60$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-11n+\left(-\frac{11}{2}\right)^{2}=60+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-11n+\frac{121}{4}=60+\frac{121}{4}$$

Add 60 to \frac{121}{4}.

$$n^{2}-11n+\frac{121}{4}=\frac{361}{4}$$

Factor n^{2}-11n+\frac{121}{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}.

$$\left(n-\frac{11}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{11}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$n-\frac{11}{2}=\frac{19}{2} n-\frac{11}{2}=-\frac{19}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$n=15 n=-4$$