How do you solve x^2+4x+1=0 by completing the square?

1 Answer
May 6, 2016

$x = - 2 \pm \sqrt{3}$

Explanation:

In addition to completing the square, use the difference of squares identity:

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

with $a = \left(x + 2\right)$ and $b = \sqrt{3}$ as follows:

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

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

$= {\left(x + 2\right)}^{2} - 3$

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

$= \left(\left(x + 2\right) - \sqrt{3}\right) \left(\left(x + 2\right) + \sqrt{3}\right)$

$= \left(x + 2 - \sqrt{3}\right) \left(x + 2 + \sqrt{3}\right)$

Hence:

$x = - 2 \pm \sqrt{3}$