Solve 6n^2-12n=90

$$6 n ^ { 2 } - 12 n = 90$$

Solve for n

$$n=-3$$

$$n=5$$

Steps Using Factoring By Grouping

Subtract 90 from both sides.

$$6n^{2}-12n-90=0$$

Divide both sides by 6.

$$n^{2}-2n-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn-15. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-15\right)=-15$$

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 -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -2.

$$a=-5 b=3$$

Rewrite n^{2}-2n-15 as \left(n^{2}-5n\right)+\left(3n-15\right).

$$\left(n^{2}-5n\right)+\left(3n-15\right)$$

Factor out n in the first and 3 in the second group.

$$n\left(n-5\right)+3\left(n-5\right)$$

Factor out common term n-5 by using distributive property.

$$\left(n-5\right)\left(n+3\right)$$

To find equation solutions, solve n-5=0 and n+3=0.

$$n=5 n=-3$$

Steps Using the Quadratic Formula

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.

$$6n^{2}-12n=90$$

Subtract 90 from both sides of the equation.

$$6n^{2}-12n-90=90-90$$

Subtracting 90 from itself leaves 0.

$$6n^{2}-12n-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -12 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\left(-90\right)}}{2\times 6}$$

Square -12.

$$n=\frac{-\left(-12\right)±\sqrt{144-4\times 6\left(-90\right)}}{2\times 6}$$

Multiply -4 times 6.

$$n=\frac{-\left(-12\right)±\sqrt{144-24\left(-90\right)}}{2\times 6}$$

Multiply -24 times -90.

$$n=\frac{-\left(-12\right)±\sqrt{144+2160}}{2\times 6}$$

Add 144 to 2160.

$$n=\frac{-\left(-12\right)±\sqrt{2304}}{2\times 6}$$

Take the square root of 2304.

$$n=\frac{-\left(-12\right)±48}{2\times 6}$$

The opposite of -12 is 12.

$$n=\frac{12±48}{2\times 6}$$

Multiply 2 times 6.

$$n=\frac{12±48}{12}$$

Now solve the equation n=\frac{12±48}{12} when ± is plus. Add 12 to 48.

$$n=\frac{60}{12}$$

Divide 60 by 12.

$$n=5$$

Now solve the equation n=\frac{12±48}{12} when ± is minus. Subtract 48 from 12.

$$n=-\frac{36}{12}$$

Divide -36 by 12.

$$n=-3$$

The equation is now solved.

$$n=5 n=-3$$

Steps for Completing the Square

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.

$$6n^{2}-12n=90$$

Divide both sides by 6.

$$\frac{6n^{2}-12n}{6}=\frac{90}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$n^{2}+\left(-\frac{12}{6}\right)n=\frac{90}{6}$$

Divide -12 by 6.

$$n^{2}-2n=\frac{90}{6}$$

Divide 90 by 6.

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

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

$$n^{2}-2n+1=15+1$$

Add 15 to 1.

$$n^{2}-2n+1=16$$

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

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

Take the square root of both sides of the equation.

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

Simplify.

$$n-1=4 n-1=-4$$

Add 1 to both sides of the equation.

$$n=5 n=-3$$