Solve 66=2n^2-n

$$66 = 2 n ^ { 2 } - n$$

Solve for n

$$n = -\frac{11}{2} = -5\frac{1}{2} = -5.5$$

$$n=6$$

Steps Using Factoring By Grouping

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

$$2n^{2}-n=66$$

Subtract 66 from both sides.

$$2n^{2}-n-66=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2n^{2}+an+bn-66. To find a and b, set up a system to be solved.

$$a+b=-1 ab=2\left(-66\right)=-132$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -132.

$$1,-132 2,-66 3,-44 4,-33 6,-22 11,-12$$

Calculate the sum for each pair.

$$1-132=-131 2-66=-64 3-44=-41 4-33=-29 6-22=-16 11-12=-1$$

The solution is the pair that gives sum -1.

$$a=-12 b=11$$

Rewrite 2n^{2}-n-66 as \left(2n^{2}-12n\right)+\left(11n-66\right).

$$\left(2n^{2}-12n\right)+\left(11n-66\right)$$

Factor out 2n in the first and 11 in the second group.

$$2n\left(n-6\right)+11\left(n-6\right)$$

Factor out common term n-6 by using distributive property.

$$\left(n-6\right)\left(2n+11\right)$$

To find equation solutions, solve n-6=0 and 2n+11=0.

$$n=6 n=-\frac{11}{2}$$

Steps Using the Quadratic Formula

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

$$2n^{2}-n=66$$

Subtract 66 from both sides.

$$2n^{2}-n-66=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -66 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-66\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-\left(-1\right)±\sqrt{1-8\left(-66\right)}}{2\times 2}$$

Multiply -8 times -66.

$$n=\frac{-\left(-1\right)±\sqrt{1+528}}{2\times 2}$$

Add 1 to 528.

$$n=\frac{-\left(-1\right)±\sqrt{529}}{2\times 2}$$

Take the square root of 529.

$$n=\frac{-\left(-1\right)±23}{2\times 2}$$

The opposite of -1 is 1.

$$n=\frac{1±23}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{1±23}{4}$$

Now solve the equation n=\frac{1±23}{4} when ± is plus. Add 1 to 23.

$$n=\frac{24}{4}$$

Divide 24 by 4.

$$n=6$$

Now solve the equation n=\frac{1±23}{4} when ± is minus. Subtract 23 from 1.

$$n=-\frac{22}{4}$$

Reduce the fraction \frac{-22}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$n=6 n=-\frac{11}{2}$$

Steps for Completing the Square

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

$$2n^{2}-n=66$$

Divide both sides by 2.

$$\frac{2n^{2}-n}{2}=\frac{66}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$n^{2}-\frac{1}{2}n=\frac{66}{2}$$

Divide 66 by 2.

$$n^{2}-\frac{1}{2}n=33$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{1}{2}n+\left(-\frac{1}{4}\right)^{2}=33+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{1}{2}n+\frac{1}{16}=33+\frac{1}{16}$$

Add 33 to \frac{1}{16}.

$$n^{2}-\frac{1}{2}n+\frac{1}{16}=\frac{529}{16}$$

Factor n^{2}-\frac{1}{2}n+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{1}{4}\right)^{2}=\frac{529}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{4}\right)^{2}}=\sqrt{\frac{529}{16}}$$

Simplify.

$$n-\frac{1}{4}=\frac{23}{4} n-\frac{1}{4}=-\frac{23}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$n=6 n=-\frac{11}{2}$$