# How do you factor x^2 -x+7?

Feb 22, 2016

Use the quadratic formula to find:

${x}^{2} - x + 7 = \left(x - \frac{1}{2} - \frac{3 \sqrt{3}}{2} i\right) \left(x - \frac{1}{2} + \frac{3 \sqrt{3}}{2} i\right)$

#### Explanation:

$f \left(x\right) = {x}^{2} - x + 7$ is in the form $a {x}^{2} + b x + c$ with $a = 1$, $b = - 1$ and $c = 7$

This has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c = {\left(- 1\right)}^{2} - \left(4 \cdot 1 \cdot 7\right) = 1 - 28 = - 27$

Since this is negative $f \left(x\right)$ has no linear factors with Real coefficients. We can find its Complex factorisation by using the quadratic formula then converting the zeros into factors:

The roots of $f \left(x\right) = 0$ are given by the quadratic formula:

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

$= \frac{- b \pm \sqrt{\Delta}}{2 a}$

$= \frac{1 \pm \sqrt{- 27}}{2}$

$= \frac{1}{2} \pm \frac{\sqrt{27}}{2} i$

$= \frac{1}{2} \pm \frac{3 \sqrt{3}}{2} i$

Hence $f \left(x\right)$ can be factored as:

${x}^{2} - x + 7 = \left(x - \frac{1}{2} - \frac{3 \sqrt{3}}{2} i\right) \left(x - \frac{1}{2} + \frac{3 \sqrt{3}}{2} i\right)$