Question

The solutions to 4x^2 + 3 = 3x - 9 can be written in the form x = a \pm b i, where a and b are real numbers. What is a + b^2? Express your answer as a fraction.

Please help quickly!

Answer 1

Complete square!

4x^2+3=3x-9
=>
4x^2-3x+12=0
4(x^2-3/4x)+12=0
4(x-3/8)^2-9/16+16=0
4(x-3/8)^2 + 183/16=0
(x-3/8)^2 =-183/64=0
x-3/8 = +/- (sqrt(183)/8) i
x=3/8 +/- sqrt(183)/8 i
=>
a=3/8, b=sqrt(183)/8
so
a+b^2
=3/8+183/64
=207/64