SOLUTION: Find a quadratic equation with roots (4+i) and (4-i).

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Question 166677: Find a quadratic equation with roots (4+i) and (4-i).
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!



Since 4%2Bi and 4-i are given zeros this means that:


x=4%2Bi and x=4-i


Get all terms to the left side in each case


x-%284%2Bi%29=0 and x-%284-i%29=0


Now use the zero product property in reverse to join the factors.


%28x-%284%2Bi%29%29%28x-%284-i%29%29=0


%28%28x-4%29%2Bi%29%28%28x-4%29-i%29%29=0 Regroup the terms.


Now let w=x-4. So the equation is now:


%28w%2Bi%29%28w-i%29=0


w%5E2-i%5E2=0 FOIL


%28x-4%29%5E2-%28-1%29=0 Plug in w=x-4 and i%5E2=-1


%28x-4%29%5E2%2B1=0 Rewrite %28x-4%29%5E2-%28-1%29 as %28x-4%29%5E2%2B1


x%5E2-8x%2B16%2B1=0 FOIL


x%5E2-8x%2B17=0 Combine like terms.



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Answer:

So the polynomial with roots of 4%2Bi and 4-i is


x%5E2-8x%2B17



Note: you can verify the answer by use of the quadratic formula.