# How do you solve the equation x^2+x+1/4=9/16 by completing the square?

Jan 27, 2017

$x = \frac{1}{4} \text{ or } - \frac{5}{4}$

#### Explanation:

rearrange as follows

${x}^{2} + x = \frac{9}{16} - \frac{1}{4}$

${x}^{2} + x = \frac{5}{16}$

complete the square $L H S$

$\left({x}^{2} + x + {\left(\frac{1}{2}\right)}^{2}\right) - {\left(\frac{1}{2}\right)}^{2} = \frac{5}{16}$

${\left(x + \frac{1}{2}\right)}^{2} - \frac{1}{4} = \frac{5}{16}$

now solve for $x$

${\left(x + \frac{1}{2}\right)}^{2} = \frac{5}{16} + \frac{1}{4} = \frac{9}{16}$

${\left(x + \frac{1}{2}\right)}^{2} = \frac{9}{16}$

$x + \frac{1}{2} = \pm \sqrt{\frac{9}{16}}$

$x + \frac{1}{2} = \pm \frac{3}{4}$

$x = - \frac{1}{2} \pm \frac{3}{4}$

${x}_{1} = - \frac{1}{2} + \frac{3}{4} = \frac{1}{4}$

${x}_{2} = - \frac{1}{2} - \frac{3}{4} = - \frac{5}{4}$