# How to factor f(x)=1/2x^2+5/2x-3/2?

## Do I have to use the quadratic formula?

Nov 11, 2016

Yes, you do have to use the quadratic formula, and after doing so, you get $f \left(x\right) = \frac{1}{8} \left(2 x + 5 - \sqrt{37}\right) \left(2 x + 5 + \sqrt{37}\right)$.

#### Explanation:

First, get rid of that pesky $\frac{1}{2}$. Fractions are usually difficult to work with, but in this case we can factor out the $\frac{1}{2}$ and not have to deal with them at all:
$f \left(x\right) = \frac{1}{2} \left({x}^{2} + 5 x - 3\right)$

Now we can ignore the $\frac{1}{2}$ and focus on the good stuff:
${x}^{2} + 5 x - 3$

We must ask ourselves, "Is there a number pair that multiplies to $- 3$ and adds to $5$?" We will quickly discover that no such pair exists (no rational numbers meet these criteria), and indeed, we must use the quadratic formula. This is due to fact that the determinant ${b}^{2} - 4 a c$ is not a perfect square.

This equation will have roots at:
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
$= \frac{- 5 \pm \sqrt{{5}^{2} - 4 \left(1\right) \left(- 3\right)}}{2 \left(1\right)}$
$= \frac{- 5 \pm \sqrt{25 + 12}}{2}$
$\to x = \frac{- 5 \pm \sqrt{37}}{2}$

If $x = a$ is a root of a polynomial, then $x - a = 0$, and $\left(x - a\right)$ is a factor of the polynomial. So if we want to factor ${x}^{2} + 5 x - 3$, we'll need the roots to look something like $x - \text{blah blah} = 0$.

Let's get to it. We have:
$x = \frac{- 5 \pm \sqrt{37}}{2}$

Doing some algebra, we can simplify this to:
$2 x + 5 = \pm \sqrt{37}$

We have two cases: $\sqrt{37}$ and $- \sqrt{37}$. We'll work with them both:
$2 x + 5 = \sqrt{37} \to 2 x + 5 - \sqrt{37} = 0$
$2 x + 5 = - \sqrt{37} \to 2 x + 5 + \sqrt{37} = 0$

So the two factors of our quadratic are $2 x + 5 - \sqrt{37} = 0$ and $2 x + 5 + \sqrt{37} = 0$, and that means we're done.

But note that multiplication of these monomials leads to $4 {x}^{2}$ not $\frac{1}{2} {x}^{2}$, so we should divide the product by $8$. Hence,

$f \left(x\right) = \frac{1}{8} \left(2 x + 5 - \sqrt{37}\right) \left(2 x + 5 + \sqrt{37}\right)$