# How do you solve the equation 2x^2+10x+1=13 by completing the square?

May 7, 2017

Put constants on one side and x terms on the other side and complete the square.

#### Explanation:

By subtracting 1 from both sides, we get:
$2 {x}^{2} + 10 x = 12$
We can simplify by dividing both sides by 2:
${x}^{2} + 5 x = 6$
Here we complete the square:
Since ${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$, here ${a}^{2}$ is ${x}^{2}$ and our $2 a b$ term is $5 x$, therefore our $b$ term must be $\frac{5}{2}$. We complete the square by making ${x}^{2} + 5 x$ into the form of the ${\left(a + b\right)}^{2}$, however we also need to subtract the ${b}^{2}$ term since we had added it in to complete the square:
$\left({x}^{2} + 5 x + {\left(\frac{5}{2}\right)}^{2}\right) - {\left(\frac{5}{2}\right)}^{2} = 6$
${\left(x + \frac{5}{2}\right)}^{2} - \frac{25}{4} = 6$
and simplify:
${\left(x + \frac{5}{2}\right)}^{2} = \frac{49}{4}$
$x + \frac{5}{2} = \pm \sqrt{\frac{49}{4}}$
$x + \frac{5}{2} = \pm \frac{7}{2}$
$x = \frac{- 5 \pm 7}{2}$
$x = - 6 \mathmr{and} x = 1$