# How do you factor y= 8m^2 - 41m - 42?

Dec 27, 2015

Find a suitable splitting of the middle term, then factor by grouping to find:

$8 {m}^{2} - 41 m - 42 = \left(8 m + 7\right) \left(m - 6\right)$

#### Explanation:

Find a pair of factors of $A C = 8 \cdot 42 = 16 \cdot 21 = 336$ which differ by $B = 41$

The split of $336$ into a pair of factors must put all of the powers of $2$ on one side, since the difference ($41$) is odd. If both factors were even, then the difference would be even too.

That leads to the following possibilities to consider:

$16 \times 21$

$\boldsymbol{48 \times 7}$

$112 \times 3$

$336 \times 1$

Having found the pair $48$, $7$ use that to split the middle term and factor by grouping:

$8 {m}^{2} - 41 m - 42$

$= 8 {m}^{2} - 48 m + 7 m - 42$

$= \left(8 {m}^{2} - 48 m\right) + \left(7 m - 42\right)$

$= 8 m \left(m - 6\right) + 7 \left(m - 6\right)$

$= \left(8 m + 7\right) \left(m - 6\right)$

Dec 27, 2015

$y = \left(8 x + 7\right) \left(x - 6\right)$

#### Explanation:

You could look for values $p , q , r , s$ such that
$\textcolor{w h i t e}{\text{XXX}} p \times r = 8$
$\textcolor{w h i t e}{\text{XXX}} q \times s = - 42$
$\textcolor{w h i t e}{\text{XXX}} p s + q r = - 41$
(perhaps using the AC method)

...but unless you get lucky, there are quite a few factorings possible.

As an alternative you could use the quadratic formula:
$\textcolor{w h i t e}{\text{XXX}} \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The numbers involved are still ugly but if you use a calculator or spreadsheet (evaluating only the $+$ of the $\pm$)
you should get:
$\textcolor{w h i t e}{\text{XXX}} x = 6$ as a zero for this expression.

Therefore one of the factors will be:
$\textcolor{w h i t e}{\text{XXX}} \left(x - 6\right)$
Simple division ($8 {x}^{2} \div x = 8$) and ((-42)div(-6))=+7)
gives the other term:
$\textcolor{w h i t e}{\text{XXX}} \left(8 x + 7\right)$

Dec 27, 2015

Alternatively, complete the square to find:

$8 {m}^{2} - 41 m - 42 = \left(m - 6\right) \left(8 m + 7\right)$

#### Explanation:

Alternatively, you can complete the square to proceed directly to the answer as follows:

$8 {m}^{2} - 41 m - 42$

$= 8 \left({m}^{2} - \frac{41}{8} m - \frac{21}{4}\right)$

$= 8 \left({m}^{2} - \frac{41}{8} m + {\left(\frac{41}{16}\right)}^{2} - {\left(\frac{41}{16}\right)}^{2} - \frac{21}{4}\right)$

$= 8 \left({\left(m - \frac{41}{16}\right)}^{2} - \frac{1681}{256} - \frac{1344}{256}\right)$

$= 8 \left({\left(m - \frac{41}{16}\right)}^{2} - \frac{3025}{256}\right)$

$= 8 \left({\left(m - \frac{41}{16}\right)}^{2} - {\left(\frac{55}{16}\right)}^{2}\right)$

$= 8 \left(\left(m - \frac{41}{16}\right) - \frac{55}{16}\right) \left(\left(m - \frac{41}{16}\right) + \frac{55}{16}\right)$

$= 8 \left(m - \frac{96}{16}\right) \left(m + \frac{14}{16}\right)$

$= 8 \left(m - 6\right) \left(m + \frac{7}{8}\right)$

$= \left(m - 6\right) \left(8 m + 7\right)$

...using the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = m - \frac{41}{16}$ and $b = \frac{55}{16}$

Ouch!