# How do you factor and solve x^2-7x+8=0?

Jan 18, 2018

$x = \frac{7 \pm \sqrt{17}}{2}$

#### Explanation:

We cannot actually factor this into simple terms, but we can use a similar method, called completing the square.

${x}^{2} - 7 x + 8 = 0$

${x}^{2} - 7 x = - 8$

Now, we need the LHS to be in the form of ${x}^{2} + 2 a x + {a}^{2} = {\left(x + a\right)}^{2}$.

In this case, $2 a x = - 7 x$

$a = - \frac{7}{2}$

Therefore, the equation becomes

${x}^{2} - 7 x + {\left(- \frac{7}{2}\right)}^{2} = - 8 + {\left(- \frac{7}{2}\right)}^{2}$

Now, we can factorize the expression

${\left(x - \frac{7}{2}\right)}^{2} = - 8 + \frac{49}{4}$

${\left(x - \frac{7}{2}\right)}^{2} = \frac{17}{4}$

$x - \frac{7}{2} = \pm \sqrt{\frac{17}{4}}$

Simplifying the right side, we get

$x - \frac{7}{2} = \frac{\pm \sqrt{17}}{2}$

Now, we can add $\frac{7}{2}$ to both sides to get,

$x = \frac{7}{2} \pm \frac{\sqrt{17}}{2}$

$x = \frac{7 \pm \sqrt{17}}{2}$