Solve x^2+15x+36

$${ x }^{ 2 } +15x+36$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+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 x^{2}+15x+36 as \left(x^{2}+3x\right)+\left(12x+36\right).

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

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

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

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

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

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.

$$x^{2}+15x+36=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\times 36}}{2}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 36}}{2}$$

Multiply -4 times 36.

$$x=\frac{-15±\sqrt{225-144}}{2}$$

Add 225 to -144.

$$x=\frac{-15±\sqrt{81}}{2}$$

Take the square root of 81.

$$x=\frac{-15±9}{2}$$

Now solve the equation x=\frac{-15±9}{2} when ± is plus. Add -15 to 9.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-15±9}{2} when ± is minus. Subtract 9 from -15.

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

Divide -24 by 2.

$$x=-12$$

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}.

$$x^{2}+15x+36=\left(x-\left(-3\right)\right)\left(x-\left(-12\right)\right)$$

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

$$x^{2}+15x+36=\left(x+3\right)\left(x+12\right)$$

Evaluate

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