How do you complete the square to solve 2k^2 + 2k = 10?

Jun 12, 2015

Divide by the coefficient of ${k}^{2}$, then add the square of half the coefficient of $k$ to both sides.

Explanation:

First, divide everything in the equation by two, to make sure that ${k}^{2}$ has a coefficient of $1$

${k}^{2} + k = 5$

Next, add ${\left(\frac{1}{2}\right)}^{2}$ to both sides of the equation

${k}^{2} + k + {\left(\frac{1}{2}\right)}^{2} = 5 + {\left(\frac{1}{2}\right)}^{2}$

The left hand side is now a perfect square

${\left(k + \frac{1}{2}\right)}^{2} = 5 + {\left(\frac{1}{2}\right)}^{2}$

Because the right hand side is positive, you can take the $\pm \sqrt{}$ of both sides

k+1/2 = +-sqrt(5 + 1/4

$k = - \frac{1}{2} \pm \sqrt{5 \frac{1}{4}} = - \frac{1}{2} \pm \sqrt{\frac{21}{4}} = - \frac{1}{2} \pm \frac{\sqrt{21}}{2}$. Finis!