Question

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

Answer 1

`d=5/6+sqrt(47)/6icolor(white)("XX")orcolor(white)("XX")d=5/6-sqrt(47)/6i`

Explanation:

For a parabolic equation of the form:
`color(white)("XXX")color(red)ad^2+color(blue)bd+color(green)c=0`
the quadratic formula gives the solution as
`color(white)("XXX")d=(-color(blue)b+-sqrt(color(blue)b^2+4color(red)acolor(green)c))/(2color(red)a)`
`color(white)("XXXXXXXXXX")`over the set of complex numbers

For the given example:
`color(white)("XXX")color(red)a=color(red)3`
`color(white)("XXX")color(blue)b=color(blue)(-5)`
`color(white)("XXX")color(green)c=color(green)6`

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

`color(white)("XXX")=(5+-sqrt(-47))/6`

`d=5/6+sqrt(47)/6icolor(white)("XX")orcolor(white)("XX")d=5/6-sqrt(47)/6i`