Factoring Quadratics

To factor a trinomial with a leading coefficient of $1,$ we need to find two numbers whose product is the independent term. $$\begin{array}{c}{x}^2-10x+24\end{array}$$ In this case, we have that the constant term is $24.$ This is a positive number, so for a product to be positive, the factors must have the same sign (both positive or both negative).

Factor Constants	Product of Constants
$1$ and $24$	$24$
$\text{-}1$ and $\text{-}24$	$24$
$2$ and $12$	$24$
$\text{-}2$ and $\text{-}12$	$24$
$3$ and $8$	$24$
$\text{-}3$ and $\text{-}8$	$24$
$4$ and $6$	$24$
$\text{-}4$ and $\text{-}6$	$24$

Next, let's consider the coefficient of the linear term. $$\begin{array}{c}{x}^2-10x+24\end{array}$$ In this case, since the linear coefficient is $\text{-}10,$ we need the sum of the factors to be $\text{-}10.$

Factors	Sum of Factors
$1$ and $24$	$25$
$\text{-}1$ and $\text{-}24$	$\text{-}25$
$2$ and $12$	$14$
$\text{-}2$ and $\text{-}12$	$\text{-}14$
$3$ and $8$	$11$
$\text{-}3$ and $\text{-}8$	$\text{-}11$
$4$ and $6$	$10$
$\text{-}4$ and $\text{-}6$	$\text{-}10$

We found two numbers whose product is $24$ and whose sum is $\text{-}10.$ These numbers are $\text{-}4$ and $\text{-}6.$ With this we can factor the given trinomial.