How do you use solve x^2 - 10x + 8 = 0?

1 Answer
Mar 3, 2016

Complete the square and use the difference of squares identity to find:

$x = 5 \pm \sqrt{17}$

Explanation:

The difference of squares identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

We use this with $a = \left(x - 5\right)$ and $b = \sqrt{17}$ below.

Complete the square then use the above as follows:

$0 = {x}^{2} - 10 x + 8$

$= {x}^{2} - 10 x + 25 - 17$

$= {\left(x - 5\right)}^{2} - {\left(\sqrt{17}\right)}^{2}$

$= \left(\left(x - 5\right) - \sqrt{17}\right) \left(\left(x - 5\right) + \sqrt{17}\right)$

$= \left(x - 5 - \sqrt{17}\right) \left(x - 5 + \sqrt{17}\right)$

Hence: $x = 5 \pm \sqrt{17}$