Solve 12+13m+m^2

$$12 + 13 m + m ^ { 2 }$$

Factor

$$\left(m+1\right)\left(m+12\right)$$

Solution Steps

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

$$m^{2}+13m+12$$

Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+12. To find a and b, set up a system to be solved.

$$a+b=13 ab=1\times 12=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 13.

$$a=1 b=12$$

Rewrite m^{2}+13m+12 as \left(m^{2}+m\right)+\left(12m+12\right).

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

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

$$m\left(m+1\right)+12\left(m+1\right)$$

Factor out common term m+1 by using distributive property.

$$\left(m+1\right)\left(m+12\right)$$

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.

$$m^{2}+13m+12=0$$

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.

$$m=\frac{-13±\sqrt{13^{2}-4\times 12}}{2}$$

Square 13.

$$m=\frac{-13±\sqrt{169-4\times 12}}{2}$$

Multiply -4 times 12.

$$m=\frac{-13±\sqrt{169-48}}{2}$$

Add 169 to -48.

$$m=\frac{-13±\sqrt{121}}{2}$$

Take the square root of 121.

$$m=\frac{-13±11}{2}$$

Now solve the equation m=\frac{-13±11}{2} when ± is plus. Add -13 to 11.

$$m=-\frac{2}{2}$$

Divide -2 by 2.

$$m=-1$$

Now solve the equation m=\frac{-13±11}{2} when ± is minus. Subtract 11 from -13.

$$m=-\frac{24}{2}$$

Divide -24 by 2.

$$m=-12$$

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

$$m^{2}+13m+12=\left(m-\left(-1\right)\right)\left(m-\left(-12\right)\right)$$

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

$$m^{2}+13m+12=\left(m+1\right)\left(m+12\right)$$

Evaluate

$$\left(m+1\right)\left(m+12\right)$$