SOLUTION: The solutions to 2x^2 - 10x + 13 = -17x^2 - 14x - 28 are a+bi and a-bi, where a and b are positive. What is a + b?

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Question 1209257: The solutions to
2x^2 - 10x + 13 = -17x^2 - 14x - 28
are a+bi and a-bi, where a and b are positive. What is a + b?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52780) About Me  (Show Source):
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

I'll relabel a+bi and a-bi as p+qi and p-qi respectively.

2x%5E2+-+10x+%2B+13+=+-17x%5E2+-+14x+-+28
rearranges to
19x%5E2%2B4x%2B41+=+0
after getting everything to one side.


Compare 19x%5E2%2B4x%2B41+=+0 with ax%5E2%2Bbx%2Bc+=+0 to get these values
a = 19, b = 4, c = 41

Plug them into the quadratic formula
x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29
to generate these roots
x+=+-2%2F19+%2B+%285%2Asqrt%2831%29%2F19%29i or x+=+-2%2F19+-+%285%2Asqrt%2831%29%2F19%29i
I'll let the student handle the scratch work.

Those roots are of the form p+qi and p-qi where
matrix%281%2C3%2Cp+=+-2%2F19%2C%22and%22%2Cq+=+5%2Asqrt%2831%29%2F19%29

So,
p%2Bq+=+-2%2F19+%2B+5%2Asqrt%2831%29%2F19

p%2Bq+=+%28-2+%2B+5%2Asqrt%2831%29%29%2F19
There's not much else we can do to simplify.