How do you solve x^2+2x-6=0 by completing the square?

Oct 22, 2016

$x = - 1 \pm \sqrt{7}$

Explanation:

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

Move the constant to the right side by adding 6 to both sides.

${x}^{2} + \textcolor{m a \ge n t a}{2} x \textcolor{w h i t e}{a a a a a a} = 6$

Divide the coefficient $\textcolor{m a \ge n t a}{2}$ of the middle term $\textcolor{m a \ge n t a}{2} x$ by $2$:

$\frac{\textcolor{m a \ge n t a}{2}}{2} = \textcolor{b l u e}{1}$

Square the $\textcolor{b l u e}{1}$ and add the result to both sides.

${\textcolor{b l u e}{1}}^{2} = \textcolor{red}{1}$

${x}^{2} + 2 x + \textcolor{red}{1} = 6 + \textcolor{red}{1}$

Factor the left side and sum the right side. Notice the $\textcolor{b l u e}{1}$ in the factored form is the same $\textcolor{b l u e}{1}$ you obtained by dividing the coefficient of the middle term by $2$

$\left(x + \textcolor{b l u e}{1}\right) \left(x + \textcolor{b l u e}{1}\right) = 7$

Express the left side as the square of the binomial.

${\left(x + \textcolor{b l u e}{1}\right)}^{2} = 7$

Square root both sides.

$\sqrt{{\left(x + 1\right)}^{2}} = \sqrt{7}$

$x + 1 = \pm \sqrt{7}$

Subtract 1 from each side.

$x = - 1 \pm \sqrt{7}$