Question

How do you describe the roots of `4x^2 - 3x + 2 = 0`?

Answer 1

A conjugate pair of Complex roots.

Explanation:

`4x^2-3x+2` is of the form `ax^2+bx+c` with `a=4`, `b=-3` and `c=2`.

This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = (-3)^2-(4*4*2) = 9-32 = -23`

Since `Delta` is negative, the quadratic equation has no Real roots. It has a pair of Complex roots, conjugate to one another, given by the quadratic formula:

`x = (-b+-sqrt(b^2-4ac))/(2a)`

`=(-b+-sqrt(Delta))/(2a)`

`=(3+-sqrt(-23))/(2*4)`

`=3/8+-sqrt(23)/8i`

where `i` is the imaginary unit.