# What is factorization of quadratic expressions?

To factorize a quadratic expression of the form $a {x}^{2} + b x + c$ you must find two numbers that add together to give the first coefficient of $x$ and multiply to give the second coefficient of $x$.
An example of this would be the equation ${x}^{2} + 5 x + 6$, which factorizes to give the expression $\left(x + 6\right) \left(x - 1\right)$
Now, one might expect the solution to include the numbers 2 and 3, as these two numbers both add together to give 5 and multiply to give 6. However, as the signs differ in the factorized equation, then the solution to the equation must be $\left(x + 6\right) \left(x - 1\right)$, as $+ 6 - 1$ gives $5$, and $6 \times 1$ yields a solution of 6.
The equation can be checked by multiplying the solutions back into the equation to give the original quadratic of ${x}^{2} + 5 x + 6$.