How do you solve the equation x^2+4x=6 by completing the square?

Feb 27, 2017

$x = - 2 \pm \sqrt{10}$

Explanation:

First subtract $6$ from both sides to get the quadratic equation into standard form:

${x}^{2} + 4 x - 6 = 0$

The difference of squares identity can be written:

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

Use this with $a = x + 2$ and $b = \sqrt{10}$ as follows:

$0 = {x}^{2} + 4 x - 6$

$\textcolor{w h i t e}{0} = {x}^{2} + 4 x + 4 - 10$

$\textcolor{w h i t e}{0} = {\left(x + 2\right)}^{2} - {\left(\sqrt{10}\right)}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(x + 2\right) - \sqrt{10}\right) \left(\left(x + 2\right) + \sqrt{10}\right)$

$\textcolor{w h i t e}{0} = \left(x + 2 - \sqrt{10}\right) \left(x + 2 + \sqrt{10}\right)$

Hence:

$x = - 2 \pm \sqrt{10}$