Solve 2m^2+30m+72

$$2 { m }^{ 2 } +30m+72$$

Factor

$$2\left(m+3\right)\left(m+12\right)$$

Solution Steps

Factor out 2.

$$2\left(m^{2}+15m+36\right)$$

Consider m^{2}+15m+36. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+36. To find a and b, set up a system to be solved.

$$a+b=15 ab=1\times 36=36$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 15.

$$a=3 b=12$$

Rewrite m^{2}+15m+36 as \left(m^{2}+3m\right)+\left(12m+36\right).

$$\left(m^{2}+3m\right)+\left(12m+36\right)$$

Factor out m in the first and 12 in the second group.

$$m\left(m+3\right)+12\left(m+3\right)$$

Factor out common term m+3 by using distributive property.

$$\left(m+3\right)\left(m+12\right)$$

Rewrite the complete factored expression.

$$2\left(m+3\right)\left(m+12\right)$$

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.

$$2m^{2}+30m+72=0$$

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.

$$m=\frac{-30±\sqrt{30^{2}-4\times 2\times 72}}{2\times 2}$$

Square 30.

$$m=\frac{-30±\sqrt{900-4\times 2\times 72}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-30±\sqrt{900-8\times 72}}{2\times 2}$$

Multiply -8 times 72.

$$m=\frac{-30±\sqrt{900-576}}{2\times 2}$$

Add 900 to -576.

$$m=\frac{-30±\sqrt{324}}{2\times 2}$$

Take the square root of 324.

$$m=\frac{-30±18}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{-30±18}{4}$$

Now solve the equation m=\frac{-30±18}{4} when ± is plus. Add -30 to 18.

$$m=-\frac{12}{4}$$

Divide -12 by 4.

$$m=-3$$

Now solve the equation m=\frac{-30±18}{4} when ± is minus. Subtract 18 from -30.

$$m=-\frac{48}{4}$$

Divide -48 by 4.

$$m=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -12 for x_{2}.

$$2m^{2}+30m+72=2\left(m-\left(-3\right)\right)\left(m-\left(-12\right)\right)$$

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

$$2m^{2}+30m+72=2\left(m+3\right)\left(m+12\right)$$

Evaluate

$$2\left(m+3\right)\left(m+12\right)$$