Question 389424
A parabola of the form y=2x squared+bx+c has complex roots of -1plus/minus 4i.  Find the values of b and c


y = 2x^2 + bx + c
roots -1 +- 4i
x1 = -1 + 4i
x2 = -1 - 4i
x1 and x2 are conjugates (opposite signs on imaginary parts)
-1 + 4i - x1 = 0 --> -x1 = 1 - 4i
-1 - 4i - x2 = 0 --> -x2 = 1 + 4i
(x + 1 - 4i)(x + 1 + 4i)
x^2 + x + 4ix + x + 1 + 4i - 4ix - 4i - 16i^2, i^2 = -1
4ix cancels out, 4i cancels out
x^2 + 2x + 1 + 16
x^2 + 2x + 17 --> that does not match up to form 2x^2 + bx + c so:
y = 2x^2 + 4x + 34 (multiplied each term by 2)
b = 4, c = 34
check:
(-1 + 4i)^2 = (-1 + 4i)(-1 + 4i) with FOIL = 1 - 8i + 16i^2 = 1 - 8i - 16
(-1 + 4i)^2 = -15 - 8i
(-1 - 4i)^2 = (-1 - 4i)(-1 - 4i) with FOIL = 1 + 8i + 16i^2 = 1 + 8i - 16
(-1 + 4i)^2 = -15 + 8i
4(-1 + 4i) = -4 + 16i
4(-1 - 4i) = -4 - 16i
y = 2(-15 - 8i) - 4 + 16i + 34 = -30 - 16i - 4 + 16i + 34 = -34 + 34 = 0, yes
y = 2(-15 + 8i) - 4 - 16i + 34 = -30 + 16i - 4 - 16i + 34 = -34 + 34 = 0, yes