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.
Please Help Quick!
+118608 Why do you need help so quickly?
Are you doing a test?
Have you left your homework till the last minute and need someone else to do it for you?
No, I'm doing an AoPs question, and I have tried to do it myself but I just don't know how to find the roots. Also, this work is due in January, and I am just bored and wondering if I can learn new things by doing a bit of work ahead of time. If you could help me with that it would be much appreciated. Thank you!
Use the Quadratic formula to find the roots
a=2 b =-3 c = 27
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) then x = 3/4 +- 3/4 * i sqrt23
squaring them results in 9/16 +18/16 sqrt23 i - 9/ 16 * 23 and 9/16 - 18/16 sqrt23 i - 9/16*23
then adding them = - 198/8 = -99/4
+128460 Call the roots m and n
By Vieta's Theorem......
ax^2 + bc + c
Sum of roots = m + n = -b/a = 3/2 square both sides m^2 + 2mn + n^2 = 9/4 (1)
Product of roots = mn = c/a = 27/2 and 2mn = 27 (2)
So......subbing (2) into ( 1) we have
m^2 + 27 + n^2 = 9/4
m^2 + n^2 = 9/4 - 27
m^2 + n^2 = [9 - 108 ] / 4 = - 99 / 4
