# How do you solve the quadratic u^2-2u+3=0 using any method?

Sep 26, 2016

$u = 1 \pm \sqrt{2} i$

#### Explanation:

The difference of squares identity can be written:

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

Use this with $a = \left(u - 1\right)$ and $b = \sqrt{2} i$ as follows:

$0 = {u}^{2} - 2 u + 3$

$\textcolor{w h i t e}{0} = {u}^{2} - 2 u + 1 + 2$

$\textcolor{w h i t e}{0} = {\left(u - 1\right)}^{2} - {\left(\sqrt{2} i\right)}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(u - 1\right) - \sqrt{2} i\right) \left(\left(u - 1\right) + \sqrt{2} i\right)$

$\textcolor{w h i t e}{0} = \left(u - 1 - \sqrt{2} i\right) \left(u - 1 + \sqrt{2} i\right)$

Hence $u = 1 \pm \sqrt{2} i$