# How do you solve by completing the square: x^2+5/2x=5?

Apr 3, 2015

To complete the square on the left side we need to add the square of half of the coefficient of the $x$ term.
(This follows from the observation ${\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2}$)

${x}^{2} + \frac{5}{2} x + {\left(\frac{5}{4}\right)}^{2} = 5 + {\left(\frac{5}{4}\right)}^{2}$

${\left(x + \frac{5}{4}\right)}^{2} = \frac{105}{16}$

Taking the square root of both sides
$x + \frac{5}{4} = \pm \frac{\sqrt{105}}{4}$

So
$x = - \frac{5 + \sqrt{105}}{4}$
or
$x = - \frac{5 - \sqrt{105}}{4}$