The quadratic 2x^2-3x+27+3x^2+5 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.
simplify the quadratic:
\(2x^2 - 3x + 27 + 3x^2 + 5 = 5x^2 - 3x + 32\)
Let the roots be x and y. From Vieta's formulas, we know that:
sum of roots = x + y = 3/5
product of roots = xy = 32/5
sum of squares of roots = x^2 + y^2 = (x + y)^2 - 2xy = (3/5)^2 - 2(32/5) = -12.44