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.
$$a+b=13$$ $$ab=1\times 36=36$$
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 $36$.
Rewrite $x^{2}+13x+36$ as $\left(x^{2}+4x\right)+\left(9x+36\right)$.
$$\left(x^{2}+4x\right)+\left(9x+36\right)$$
Factor out $x$ in the first and $9$ in the second group.
$$x\left(x+4\right)+9\left(x+4\right)$$
Factor out common term $x+4$ by using distributive property.
$$\left(x+4\right)\left(x+9\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}+13x+36=0$$
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.
$$x=\frac{-13±\sqrt{13^{2}-4\times 36}}{2}$$
Square $13$.
$$x=\frac{-13±\sqrt{169-4\times 36}}{2}$$
Multiply $-4$ times $36$.
$$x=\frac{-13±\sqrt{169-144}}{2}$$
Add $169$ to $-144$.
$$x=\frac{-13±\sqrt{25}}{2}$$
Take the square root of $25$.
$$x=\frac{-13±5}{2}$$
Now solve the equation $x=\frac{-13±5}{2}$ when $±$ is plus. Add $-13$ to $5$.
$$x=-\frac{8}{2}$$
Divide $-8$ by $2$.
$$x=-4$$
Now solve the equation $x=\frac{-13±5}{2}$ when $±$ is minus. Subtract $5$ from $-13$.
$$x=-\frac{18}{2}$$
Divide $-18$ by $2$.
$$x=-9$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-4$ for $x_{1}$ and $-9$ for $x_{2}$.
Simplify all the expressions of the form $p-\left(-q\right)$ to $p+q$.
$$x^{2}+13x+36=\left(x+4\right)\left(x+9\right)$$
Steps Using Direct Factoring Method
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form $x^2+Bx+C=0$.
$$x ^ 2 +13x +36 = 0$$
Let $r$ and $s$ be the factors for the quadratic equation such that $x^2+Bx+C=(x−r)(x−s)$ where sum of factors $(r+s)=−B$ and the product of factors $rs = C$
$$r + s = -13 $$ $$ rs = 36$$
Two numbers $r$ and $s$ sum up to $-13$ exactly when the average of the two numbers is $\frac{1}{2}*-13 = -\frac{13}{2}$. You can also see that the midpoint of $r$ and $s$ corresponds to the axis of symmetry of the parabola represented by the quadratic equation $y=x^2+Bx+C$. The values of $r$ and $s$ are equidistant from the center by an unknown quantity $u$. Express $r$ and $s$ with respect to variable $u$.
