Solve -5=5/6n+n(n-1)/2*-1/6

$$- 5 = \frac { 5 } { 6 } n + \frac { n ( n - 1 ) } { 2 } \cdot \frac { - 1 } { 6 }$$

Solve for n

$$n=-4$$

$$n=15$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 6,2.

$$-30=5n+3n\left(n-1\right)\times \frac{-1}{6}$$

Fraction \frac{-1}{6} can be rewritten as -\frac{1}{6} by extracting the negative sign.

$$-30=5n+3n\left(n-1\right)\left(-\frac{1}{6}\right)$$

Express 3\left(-\frac{1}{6}\right) as a single fraction.

$$-30=5n+\frac{3\left(-1\right)}{6}n\left(n-1\right)$$

Multiply 3 and -1 to get -3.

$$-30=5n+\frac{-3}{6}n\left(n-1\right)$$

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

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

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

$$-30=5n-\frac{1}{2}nn-\frac{1}{2}n\left(-1\right)$$

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

$$-30=5n-\frac{1}{2}n^{2}-\frac{1}{2}n\left(-1\right)$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

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

Combine 5n and \frac{1}{2}n to get \frac{11}{2}n.

$$-30=\frac{11}{2}n-\frac{1}{2}n^{2}$$

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

$$\frac{11}{2}n-\frac{1}{2}n^{2}=-30$$

Add 30 to both sides.

$$\frac{11}{2}n-\frac{1}{2}n^{2}+30=0$$

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.

$$-\frac{1}{2}n^{2}+\frac{11}{2}n+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, \frac{11}{2} for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\frac{11}{2}±\sqrt{\left(\frac{11}{2}\right)^{2}-4\left(-\frac{1}{2}\right)\times 30}}{2\left(-\frac{1}{2}\right)}$$

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

$$n=\frac{-\frac{11}{2}±\sqrt{\frac{121}{4}-4\left(-\frac{1}{2}\right)\times 30}}{2\left(-\frac{1}{2}\right)}$$

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

$$n=\frac{-\frac{11}{2}±\sqrt{\frac{121}{4}+2\times 30}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times 30.

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

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

$$n=\frac{-\frac{11}{2}±\sqrt{\frac{361}{4}}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of \frac{361}{4}.

$$n=\frac{-\frac{11}{2}±\frac{19}{2}}{2\left(-\frac{1}{2}\right)}$$

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

$$n=\frac{-\frac{11}{2}±\frac{19}{2}}{-1}$$

Now solve the equation n=\frac{-\frac{11}{2}±\frac{19}{2}}{-1} when ± is plus. Add -\frac{11}{2} to \frac{19}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n=\frac{4}{-1}$$

Divide 4 by -1.

$$n=-4$$

Now solve the equation n=\frac{-\frac{11}{2}±\frac{19}{2}}{-1} when ± is minus. Subtract \frac{19}{2} from -\frac{11}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$n=-\frac{15}{-1}$$

Divide -15 by -1.

$$n=15$$

The equation is now solved.

$$n=-4 n=15$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 6,2.

$$-30=5n+3n\left(n-1\right)\times \frac{-1}{6}$$

Fraction \frac{-1}{6} can be rewritten as -\frac{1}{6} by extracting the negative sign.

$$-30=5n+3n\left(n-1\right)\left(-\frac{1}{6}\right)$$

Express 3\left(-\frac{1}{6}\right) as a single fraction.

$$-30=5n+\frac{3\left(-1\right)}{6}n\left(n-1\right)$$

Multiply 3 and -1 to get -3.

$$-30=5n+\frac{-3}{6}n\left(n-1\right)$$

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

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

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

$$-30=5n-\frac{1}{2}nn-\frac{1}{2}n\left(-1\right)$$

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

$$-30=5n-\frac{1}{2}n^{2}-\frac{1}{2}n\left(-1\right)$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

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

Combine 5n and \frac{1}{2}n to get \frac{11}{2}n.

$$-30=\frac{11}{2}n-\frac{1}{2}n^{2}$$

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

$$\frac{11}{2}n-\frac{1}{2}n^{2}=-30$$

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.

$$-\frac{1}{2}n^{2}+\frac{11}{2}n=-30$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}n^{2}+\frac{11}{2}n}{-\frac{1}{2}}=-\frac{30}{-\frac{1}{2}}$$

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

$$n^{2}+\frac{\frac{11}{2}}{-\frac{1}{2}}n=-\frac{30}{-\frac{1}{2}}$$

Divide \frac{11}{2} by -\frac{1}{2} by multiplying \frac{11}{2} by the reciprocal of -\frac{1}{2}.

$$n^{2}-11n=-\frac{30}{-\frac{1}{2}}$$

Divide -30 by -\frac{1}{2} by multiplying -30 by the reciprocal of -\frac{1}{2}.

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

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.

$$n=15 n=-4$$