# How do you factor 5x^2-22x+8?

Aug 9, 2015

Factor: y = 5x^2 - 22x + 8

Ans: (5x - 2)(x - 4)

#### Explanation:

Use the new AC Method to factor trinomial (Socratic Search):
$y = 5 {x}^{2} - 22 x + 8 =$ 5(x + p)(x + q)
Converted trinomial:
$y ' = {x}^{2} - 22 x + 40 =$ (x + p')(x + q')
Find p' and q', that have same sign, knowing sum (b = -22) and product (ac = 40). They are: p' = - 2 and q' = - 20
Therefor: $p = \frac{p '}{a} = - \frac{2}{5}$ and $q = \frac{q '}{a} = - \frac{20}{5} = - 4$
Factored form of y:
$y = 5 \left(x - \frac{2}{5}\right) \left(x - 4\right) = \left(5 x - 2\right) \left(x - 4\right)$

Sep 11, 2017

$\left(x - 4\right) \left(5 x - 2\right)$

#### Explanation:

$5 {x}^{2} \textcolor{b l u e}{- 22} x \textcolor{red}{+} 8$

Find factors of $5 \mathmr{and} 8$ whose products $\textcolor{red}{\text{ADD}}$ to $\textcolor{b l u e}{22}$

$5$ is a prime number, so the factors are only $1 \mathmr{and} 5$

The factors of $8$ are $1 \times 8$ or the other pair $2 \times 4$

It may take a bit of trial and error, but notice that $5 \times 4 = 20$ which is close to the $20$ we want.

$\text{ "5 and 8" }$ write the factors and cross multiply
$\text{ "darr" } \downarrow$
$\text{ "1color(white)(xxxx)4" } \rightarrow 5 \times 4 = 20$
$\text{ "5color(white)(xxxx)2" } \rightarrow 1 \times 2 = \underline{2}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times x} \textcolor{b l u e}{22} \text{ } \leftarrow$ the combination is correct.

The positive sign of $\textcolor{red}{+} 8$ indicates the following:

ADD the factors and the signs will be the SAME

The negative sign of $\textcolor{b l u e}{- 22}$ shows what the signs are.

The factors are $\left(1 x - 4\right) \left(5 x - 2\right)$