# How do you solve 4c^2 + 10c = -7 by completing the square?

Mar 20, 2018

$c = - \frac{5}{4} \pm \frac{\sqrt{- 3}}{4}$

#### Explanation:

1. Factor out the coefficient of ${x}^{2}$, then work with the quadratic expression which has a coefficient of ${x}^{2}$ equal to $1$
2. Check the coefficient of $x$ in the new quadratic expression and take half of it - this is the number that goes into the complete square bracket
3. Balance the constant term by subtracting the square of the number from step 2 and putting in the constant from the quadratic expression
4. The rest is arithmetic that may often involve fractions

Re-write the equations as

$4 {c}^{2} + 10 c + 7 = 0$

$4 \left({c}^{2} + \frac{10 c}{4} + \frac{7}{4}\right) = 0$

$4 \left[{\left(c + \frac{10}{4 \cdot 2}\right)}^{2} - {\left(\frac{5}{4}\right)}^{2} + \frac{7}{4}\right] = 0$

$4 \left[{\left(c + \frac{5}{4}\right)}^{2} - \frac{25}{16} + \frac{7}{4}\right] = 0$

$4 \left[{\left(c + \frac{5}{4}\right)}^{2} + \frac{3}{16}\right] = 0$

$4 {\left(c + \frac{5}{4}\right)}^{2} = - \frac{3}{16} \cdot 4$

$2 \left(c + \frac{5}{4}\right) = \pm \sqrt{- \frac{3}{4}}$

$c = - \frac{5}{4} \pm \frac{\sqrt{- 3}}{4}$

This just presents the quadratic in an alternative format.