# How do you solve using the completing the square method x^ 2 + 2x + 5 = 0?

Sep 9, 2016

$x = - 1 - 2 i$ or $x = - 1 + 2 i$

#### Explanation:

${x}^{2} + 2 x + 5 = 0$

$\Leftrightarrow {x}^{2} + 2 x + 1 + 4 = 0$

or ${x}^{2} + 2 \times x \times 1 + {1}^{2} - \left(- 4\right) = 0$

or ${\left(x + 1\right)}^{2} - {\left(2 i\right)}^{2} = 0$

Now using identity ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$, this becomes

$\left(x + 1 + 2 i\right) \left(x + 1 - 2 i\right) = 0$

i.e either $x + 1 + 2 i = 0$ and $x = - 1 - 2 i$

or $x + 1 - 2 i = 0$ and $x = - 1 + 2 i$