# How do you solve the quadratic using the quadratic formula given 3d^2-5d+6=0 over the set of complex numbers?

Oct 11, 2016

$d = \frac{5}{6} + \frac{\sqrt{47}}{6} i \textcolor{w h i t e}{\text{XX")orcolor(white)("XX}} d = \frac{5}{6} - \frac{\sqrt{47}}{6} i$

#### Explanation:

For a parabolic equation of the form:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} {d}^{2} + \textcolor{b l u e}{b} d + \textcolor{g r e e n}{c} = 0$
the quadratic formula gives the solution as
$\textcolor{w h i t e}{\text{XXX}} d = \frac{- \textcolor{b l u e}{b} \pm \sqrt{{\textcolor{b l u e}{b}}^{2} + 4 \textcolor{red}{a} \textcolor{g r e e n}{c}}}{2 \textcolor{red}{a}}$
$\textcolor{w h i t e}{\text{XXXXXXXXXX}}$over the set of complex numbers

For the given example:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} = \textcolor{red}{3}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{b} = \textcolor{b l u e}{- 5}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{c} = \textcolor{g r e e n}{6}$

Therefore
color(white)("XXX")d=(-color(blue)(""(-5))+-sqrt(color(blue)(""(-5))^2-4xxcolor(red)3xxcolor(green)6))/(2xxcolor(red)3)

$\textcolor{w h i t e}{\text{XXX}} = \frac{5 \pm \sqrt{- 47}}{6}$

$d = \frac{5}{6} + \frac{\sqrt{47}}{6} i \textcolor{w h i t e}{\text{XX")orcolor(white)("XX}} d = \frac{5}{6} - \frac{\sqrt{47}}{6} i$