Consider $u^{2}+12u+35$. Factor the expression by grouping. First, the expression needs to be rewritten as $u^{2}+au+bu+35$. To find $a$ and $b$, set up a system to be solved.
$$a+b=12$$ $$ab=1\times 35=35$$
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 $35$.
$$1,35$$ $$5,7$$
Calculate the sum for each pair.
$$1+35=36$$ $$5+7=12$$
The solution is the pair that gives sum $12$.
$$a=5$$ $$b=7$$
Rewrite $u^{2}+12u+35$ as $\left(u^{2}+5u\right)+\left(7u+35\right)$.
$$\left(u^{2}+5u\right)+\left(7u+35\right)$$
Factor out $u$ in the first and $7$ in the second group.
$$u\left(u+5\right)+7\left(u+5\right)$$
Factor out common term $u+5$ by using distributive property.
$$\left(u+5\right)\left(u+7\right)$$
Rewrite the complete factored expression.
$$3\left(u+5\right)\left(u+7\right)$$
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 $ax^{2}+bx+c=0$.
$$3u^{2}+36u+105=0$$
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.
