Solve 7n^2=-8+15n

$$7 n ^ { 2 } = - 8 + 15 n$$

Solve for n

$$n=1$$

$$n = \frac{8}{7} = 1\frac{1}{7} \approx 1.142857143$$

Steps Using Factoring By Grouping

Subtract -8 from both sides.

$$7n^{2}-\left(-8\right)=15n$$

The opposite of -8 is 8.

$$7n^{2}+8=15n$$

Subtract 15n from both sides.

$$7n^{2}+8-15n=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$7n^{2}-15n+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7n^{2}+an+bn+8. To find a and b, set up a system to be solved.

$$a+b=-15 ab=7\times 8=56$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 56.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Calculate the sum for each pair.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

The solution is the pair that gives sum -15.

$$a=-8 b=-7$$

Rewrite 7n^{2}-15n+8 as \left(7n^{2}-8n\right)+\left(-7n+8\right).

$$\left(7n^{2}-8n\right)+\left(-7n+8\right)$$

Factor out n in the first and -1 in the second group.

$$n\left(7n-8\right)-\left(7n-8\right)$$

Factor out common term 7n-8 by using distributive property.

$$\left(7n-8\right)\left(n-1\right)$$

To find equation solutions, solve 7n-8=0 and n-1=0.

$$n=\frac{8}{7} n=1$$

Steps Using the Quadratic Formula

Subtract -8 from both sides.

$$7n^{2}-\left(-8\right)=15n$$

The opposite of -8 is 8.

$$7n^{2}+8=15n$$

Subtract 15n from both sides.

$$7n^{2}+8-15n=0$$

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.

$$7n^{2}-15n+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -15 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 7\times 8}}{2\times 7}$$

Square -15.

$$n=\frac{-\left(-15\right)±\sqrt{225-4\times 7\times 8}}{2\times 7}$$

Multiply -4 times 7.

$$n=\frac{-\left(-15\right)±\sqrt{225-28\times 8}}{2\times 7}$$

Multiply -28 times 8.

$$n=\frac{-\left(-15\right)±\sqrt{225-224}}{2\times 7}$$

Add 225 to -224.

$$n=\frac{-\left(-15\right)±\sqrt{1}}{2\times 7}$$

Take the square root of 1.

$$n=\frac{-\left(-15\right)±1}{2\times 7}$$

The opposite of -15 is 15.

$$n=\frac{15±1}{2\times 7}$$

Multiply 2 times 7.

$$n=\frac{15±1}{14}$$

Now solve the equation n=\frac{15±1}{14} when ± is plus. Add 15 to 1.

$$n=\frac{16}{14}$$

Reduce the fraction \frac{16}{14} to lowest terms by extracting and canceling out 2.

$$n=\frac{8}{7}$$

Now solve the equation n=\frac{15±1}{14} when ± is minus. Subtract 1 from 15.

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

Divide 14 by 14.

$$n=1$$

The equation is now solved.

$$n=\frac{8}{7} n=1$$

Steps for Completing the Square

Subtract 15n from both sides.

$$7n^{2}-15n=-8$$

Divide both sides by 7.

$$\frac{7n^{2}-15n}{7}=-\frac{8}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$n^{2}-\frac{15}{7}n=-\frac{8}{7}$$

Divide -\frac{15}{7}, the coefficient of the x term, by 2 to get -\frac{15}{14}. Then add the square of -\frac{15}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{15}{7}n+\left(-\frac{15}{14}\right)^{2}=-\frac{8}{7}+\left(-\frac{15}{14}\right)^{2}$$

Square -\frac{15}{14} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{15}{7}n+\frac{225}{196}=-\frac{8}{7}+\frac{225}{196}$$

Add -\frac{8}{7} to \frac{225}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{15}{7}n+\frac{225}{196}=\frac{1}{196}$$

Factor n^{2}-\frac{15}{7}n+\frac{225}{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}.

$$\left(n-\frac{15}{14}\right)^{2}=\frac{1}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{15}{14}\right)^{2}}=\sqrt{\frac{1}{196}}$$

Simplify.

$$n-\frac{15}{14}=\frac{1}{14} n-\frac{15}{14}=-\frac{1}{14}$$

Add \frac{15}{14} to both sides of the equation.

$$n=\frac{8}{7} n=1$$