The quadratic $2x^2-3x+27$ has two imaginary roots. What is the sum of the squares of these roots? Express your answer as a decimal rounded to the nearest hundredth.
Roots are 3/4 ± i 3/4√(23)
squared and added together = - 24.75
Call the roots a, b
The sumof the roots = a + b = 3/2
Square this
a^2 + 2ab + b^2 = 9/4
The product of the roots = ab = 27/2 ⇒ 2ab = 27
So
a^2 + 27 + b^2 = 9/4
a^2 + b^2 = 9/4 - 27 = 2.25 - 27 = -24.75