SOLUTION: find a quadratic equation with roots -1+4i and -1-4i

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Question 490686: find a quadratic equation with roots -1+4i and -1-4i
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

if you have the roots x1+=+-1%2B4i and x2+=+-1-4i you can recover the quadratic equation by forming

%28x+-+%28-1%2B4i%29%29%28x+-%28+-1-4i%29%29=0
%28x+%2B1-4i%29%28x+%2B1%2B4i%29+=+0
x%5E2%2Bx%2B4ix%2Bx+%2B1%2B4i-4ix-4i-%284i%29%5E2=0



x%5E2%2Bx%2Bx+%2B1-%2816%28-1%29%29=0

x%5E2%2B2x%2B1-%28-16%29=0

x%5E2%2B2x%2B1%2B16=0

x%5E2%2B2x%2B17=0.......your answer
check it:
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


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




So lets solve x%5E2%2B2%2Ax%2B17=0 ( notice a=1, b=2, and c=17)





x+=+%28-2+%2B-+sqrt%28+%282%29%5E2-4%2A1%2A17+%29%29%2F%282%2A1%29 Plug in a=1, b=2, and c=17




x+=+%28-2+%2B-+sqrt%28+4-4%2A1%2A17+%29%29%2F%282%2A1%29 Square 2 to get 4




x+=+%28-2+%2B-+sqrt%28+4%2B-68+%29%29%2F%282%2A1%29 Multiply -4%2A17%2A1 to get -68




x+=+%28-2+%2B-+sqrt%28+-64+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-2+%2B-+8%2Ai%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-2+%2B-+8%2Ai%29%2F%282%29 Multiply 2 and 1 to get 2




After simplifying, the quadratic has roots of


x=-1%2B4%2Ai or x=-1-4%2Ai