# Question #a3836

Jan 6, 2018

$- 2 \left(x + \frac{11 + \sqrt{329}}{4}\right) \left(x - \frac{\sqrt{329} - 11}{4}\right)$, but we will say that it's not factorable.

#### Explanation:

$26 - 11 x - 2 {x}^{2} = - 2 {x}^{2} - 11 x + 26$

$- 2 {x}^{2} - 11 x + 26$ is not factorable.

The roots of $- 2 {x}^{2} - 11 x + 26$ are $x = - \frac{11 + \sqrt{329}}{4}$ and $x = \frac{\sqrt{329} - 11}{4}$.

$\left(x + \frac{11 + \sqrt{329}}{4}\right) \left(x - \frac{\sqrt{329} - 11}{4}\right) = {x}^{2} + \frac{11 x}{2} - 13$

$\therefore - 2 \left({x}^{2} + \frac{11 x}{2} - 13\right) = - 2 {x}^{2} - 11 x + 26 = - 2 \left(x + \frac{11 + \sqrt{329}}{4}\right) \left(x - \frac{\sqrt{329} - 11}{4}\right)$

Jan 6, 2018

Q: Factorize: $y = - 2 {x}^{2} - 11 x + 26$

A: $y = - 2 {\left(x + \frac{11}{4}\right)}^{2} + \frac{329}{8}$

#### Explanation:

Factorise by completing the square;
$y = - 2 {x}^{2} - 11 x + 26$
1) Convert to a monic quadratic function by dividing both sides of the equality by $- 2$
$\frac{y}{- 2} = \frac{- 2 {x}^{2} - 11 x + 26}{- 2}$

$- \frac{y}{2} = {x}^{2} + \frac{11 x}{2} - 13$

2) Add 13 to both sides
$- \frac{y}{2} + 13 = {x}^{2} + \frac{11 x}{2}$

3) Add the square of half of the $x$ term $\frac{11}{2}$ (which is $\frac{11}{4}$) to both sides
$- \frac{y}{2} + 13 + {\left(\frac{11}{4}\right)}^{2} = {x}^{2} + \frac{11 x}{2} + {\left(\frac{11}{4}\right)}^{2}$

4) Combine the like terms on the left of the equality
$- \frac{y}{2} + \frac{329}{16} = {x}^{2} + \frac{11 x}{2} + {\left(\frac{11}{4}\right)}^{2}$

5) What we have been trying to do is create a perfect square on the right hand side of the equality. Now it is time to factorise that perfect square.

$- \frac{y}{2} + \frac{329}{16} = {\left(x + \frac{11}{4}\right)}^{2}$

6) Now we are going to move all of the stuff on the left of the equality back to the right (barring the $y$ variable of course)

$- \frac{y}{2} = {\left(x + \frac{11}{4}\right)}^{2} - \frac{329}{16}$

$y = - 2 \left[{\left(x + \frac{11}{4}\right)}^{2} - \frac{329}{16}\right]$

7) Multiplying the $- 2$ into the brackets will give us our final answer

$y = - 2 {\left(x + \frac{11}{4}\right)}^{2} + \frac{329}{8}$

I hope that helps :)

Harold