# Is it possible to factor y=-8x^2 +8x+32? If so, what are the factors?

Mar 13, 2018

$- 8$ and $\left({x}^{2} - x - 4\right)$
Solutions to the quadratic requires use of the quadratic formula (or, equivalently, completion of the square)

#### Explanation:

It is possible to pull out a common factor of $8$ (or $- 8$) from the outset to yield:

$- 8 \left({x}^{2} - x - 4\right)$

I chose to pull out $- 8$ to leave the coefficient of the term in ${x}^{2}$ as positive for convenience.

The question then arises whether there are any further factorisations of the remaining quadratic in the brackets.

Comparing with the form

$a {x}^{2} + b x + c$

It might be seen that
$a$ corresponds to $1$
$b$ corresponds to $- 1$
$c$ corresponds to $- 4$

These can be used to evaluate the "discriminant" part of the quadratic formula, which is the part:

$\sqrt{{b}^{2} - 4 a c}$

which evaluates to

$\sqrt{{\left(- 1\right)}^{2} - 4 \left(1\right) \left(- 4\right)}$

$= \sqrt{1 + 16}$

$= \sqrt{17}$

As 17 is not a perfect square, there are no further "neat" whole number factorisations of the quadratic. Non-integer (in fact non-rational but not non-real) roots may, of course, be found using the quadratic formula.

For completeness, these are

$\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$= \frac{- \left(- 1\right) \pm \sqrt{17}}{2 \left(- 1\right)}$

that is

$x = - \frac{1}{2} \pm \frac{\sqrt{17}}{2}$

Mar 13, 2018

$- 8 {x}^{2} + 8 x + 32 = - 2 \left(2 x - 1 - \sqrt{17}\right) \left(2 x - 1 + \sqrt{17}\right)$

#### Explanation:

Given:

$y = - 8 {x}^{2} + 8 x + 32$

We can factor this by completing the square and using the difference of squares identity:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

with $A = \left(2 x - 1\right)$ and $B = \sqrt{17}$ as follows:

$y = - 8 {x}^{2} + 8 x + 32$

$\textcolor{w h i t e}{y} = - 2 \left(4 {x}^{2} - 4 x + 1 - 17\right)$

$\textcolor{w h i t e}{y} = - 2 \left({\left(2 x - 1\right)}^{2} - {\left(\sqrt{17}\right)}^{2}\right)$

$\textcolor{w h i t e}{y} = - 2 \left(\left(2 x - 1\right) - \sqrt{17}\right) \left(\left(2 x - 1\right) + \sqrt{17}\right)$

$\textcolor{w h i t e}{y} = - 2 \left(2 x - 1 - \sqrt{17}\right) \left(2 x - 1 + \sqrt{17}\right)$