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

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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
Found 2 solutions by ewatrrr, CharlesG2:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

y^2 + bx + c with roots -1 ± 4i

Referring to the Quadratic formula

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

b = 1

sqrt%28+b%5E2-4%2Aa%2Ac+%29+=+4i

thus

b^2 - 4ac = -16

 1 -4*1*c = -16

   17 = 4c

   17/4 = c

Answer by CharlesG2(834) About Me  (Show Source):
You can put this solution on YOUR website!
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