Solve (2n^2+6n-108)

$$( 2 n ^ { 2 } + 6 n - 108 )$$

Factor

$$2\left(n-6\right)\left(n+9\right)$$

Solution Steps

Factor out 2.

$$2\left(n^{2}+3n-54\right)$$

Consider n^{2}+3n-54. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn-54. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\left(-54\right)=-54$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -54.

$$-1,54 -2,27 -3,18 -6,9$$

Calculate the sum for each pair.

$$-1+54=53 -2+27=25 -3+18=15 -6+9=3$$

The solution is the pair that gives sum 3.

$$a=-6 b=9$$

Rewrite n^{2}+3n-54 as \left(n^{2}-6n\right)+\left(9n-54\right).

$$\left(n^{2}-6n\right)+\left(9n-54\right)$$

Factor out n in the first and 9 in the second group.

$$n\left(n-6\right)+9\left(n-6\right)$$

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

$$\left(n-6\right)\left(n+9\right)$$

Rewrite the complete factored expression.

$$2\left(n-6\right)\left(n+9\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$2n^{2}+6n-108=0$$

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.

$$n=\frac{-6±\sqrt{6^{2}-4\times 2\left(-108\right)}}{2\times 2}$$

Square 6.

$$n=\frac{-6±\sqrt{36-4\times 2\left(-108\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-6±\sqrt{36-8\left(-108\right)}}{2\times 2}$$

Multiply -8 times -108.

$$n=\frac{-6±\sqrt{36+864}}{2\times 2}$$

Add 36 to 864.

$$n=\frac{-6±\sqrt{900}}{2\times 2}$$

Take the square root of 900.

$$n=\frac{-6±30}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-6±30}{4}$$

Now solve the equation n=\frac{-6±30}{4} when ± is plus. Add -6 to 30.

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

Divide 24 by 4.

$$n=6$$

Now solve the equation n=\frac{-6±30}{4} when ± is minus. Subtract 30 from -6.

$$n=-\frac{36}{4}$$

Divide -36 by 4.

$$n=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -9 for x_{2}.

$$2n^{2}+6n-108=2\left(n-6\right)\left(n-\left(-9\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$2n^{2}+6n-108=2\left(n-6\right)\left(n+9\right)$$

Evaluate

$$2\left(n-6\right)\left(n+9\right)$$