# How do you factor the expression 4s^2-22s+10?

Jan 19, 2017

$4 \left(x - 5\right) \left(x - \frac{1}{2}\right)$

#### Explanation:

We need to find the roots of the equation:

$4 {s}^{2} - 22 s + 10 = 0$.

We can first divide the entire expression by $2$:

$2 {s}^{2} - 11 s + 5 = 0$. Although not necessary, this does simplify the procedure just a little bit.

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$, where

$a$ is the coefficient of ${x}^{2}$
$b$ is the coefficient of $x$, and
$c$ is the constant term.

So, $x = \frac{11 \pm \sqrt{81}}{4} \implies x = 5$ or $x = \frac{1}{2}$.

The formula for factoring a quadratic is:

$a \left(x - {r}_{1}\right) \left(x - {r}_{2}\right)$ where ${r}_{1} , {r}_{2}$ the two roots.

We still have to remember that although the roots of

$4 {s}^{2} - 22 s + 10 = 0$

are the same as those of

$2 {s}^{2} - 11 s + 5 = 0$,

the original value for $a$ is $4$, not $2$.

So,

$4 {s}^{2} - 22 s + 10 = 4 \left(x - 5\right) \left(x - \frac{1}{2}\right)$