Solve 0=7n^2-3n-972

$$0 = 7 n ^ { 2 } - 3 n - 972$$

Solve for n

$$n = -\frac{81}{7} = -11\frac{4}{7} \approx -11.571428571$$

$$n=12$$

Steps Using Factoring By Grouping

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

$$7n^{2}-3n-972=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7n^{2}+an+bn-972. To find a and b, set up a system to be solved.

$$a+b=-3 ab=7\left(-972\right)=-6804$$

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 -6804.

$$1,-6804 2,-3402 3,-2268 4,-1701 6,-1134 7,-972 9,-756 12,-567 14,-486 18,-378 21,-324 27,-252 28,-243 36,-189 42,-162 54,-126 63,-108 81,-84$$

Calculate the sum for each pair.

$$1-6804=-6803 2-3402=-3400 3-2268=-2265 4-1701=-1697 6-1134=-1128 7-972=-965 9-756=-747 12-567=-555 14-486=-472 18-378=-360 21-324=-303 27-252=-225 28-243=-215 36-189=-153 42-162=-120 54-126=-72 63-108=-45 81-84=-3$$

The solution is the pair that gives sum -3.

$$a=-84 b=81$$

Rewrite 7n^{2}-3n-972 as \left(7n^{2}-84n\right)+\left(81n-972\right).

$$\left(7n^{2}-84n\right)+\left(81n-972\right)$$

Factor out 7n in the first and 81 in the second group.

$$7n\left(n-12\right)+81\left(n-12\right)$$

Factor out common term n-12 by using distributive property.

$$\left(n-12\right)\left(7n+81\right)$$

To find equation solutions, solve n-12=0 and 7n+81=0.

$$n=12 n=-\frac{81}{7}$$

Steps Using the Quadratic Formula

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

$$7n^{2}-3n-972=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -3 for b, and -972 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 7\left(-972\right)}}{2\times 7}$$

Square -3.

$$n=\frac{-\left(-3\right)±\sqrt{9-4\times 7\left(-972\right)}}{2\times 7}$$

Multiply -4 times 7.

$$n=\frac{-\left(-3\right)±\sqrt{9-28\left(-972\right)}}{2\times 7}$$

Multiply -28 times -972.

$$n=\frac{-\left(-3\right)±\sqrt{9+27216}}{2\times 7}$$

Add 9 to 27216.

$$n=\frac{-\left(-3\right)±\sqrt{27225}}{2\times 7}$$

Take the square root of 27225.

$$n=\frac{-\left(-3\right)±165}{2\times 7}$$

The opposite of -3 is 3.

$$n=\frac{3±165}{2\times 7}$$

Multiply 2 times 7.

$$n=\frac{3±165}{14}$$

Now solve the equation n=\frac{3±165}{14} when ± is plus. Add 3 to 165.

$$n=\frac{168}{14}$$

Divide 168 by 14.

$$n=12$$

Now solve the equation n=\frac{3±165}{14} when ± is minus. Subtract 165 from 3.

$$n=-\frac{162}{14}$$

Reduce the fraction \frac{-162}{14} to lowest terms by extracting and canceling out 2.

$$n=-\frac{81}{7}$$

The equation is now solved.

$$n=12 n=-\frac{81}{7}$$

Steps for Completing the Square

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

$$7n^{2}-3n-972=0$$

Add 972 to both sides. Anything plus zero gives itself.

$$7n^{2}-3n=972$$

Divide both sides by 7.

$$\frac{7n^{2}-3n}{7}=\frac{972}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$n^{2}-\frac{3}{7}n=\frac{972}{7}$$

Divide -\frac{3}{7}, the coefficient of the x term, by 2 to get -\frac{3}{14}. Then add the square of -\frac{3}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{3}{7}n+\left(-\frac{3}{14}\right)^{2}=\frac{972}{7}+\left(-\frac{3}{14}\right)^{2}$$

Square -\frac{3}{14} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{3}{7}n+\frac{9}{196}=\frac{972}{7}+\frac{9}{196}$$

Add \frac{972}{7} to \frac{9}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{3}{7}n+\frac{9}{196}=\frac{27225}{196}$$

Factor n^{2}-\frac{3}{7}n+\frac{9}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{3}{14}\right)^{2}=\frac{27225}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{3}{14}\right)^{2}}=\sqrt{\frac{27225}{196}}$$

Simplify.

$$n-\frac{3}{14}=\frac{165}{14} n-\frac{3}{14}=-\frac{165}{14}$$

Add \frac{3}{14} to both sides of the equation.

$$n=12 n=-\frac{81}{7}$$