Complex Root

Algebra Level 3

If one of the roots of the quadratic equation x 2 + b x + c = 0 x^2+bx+c=0 is 3 + 2 i 3+2i , then what is the value of ( b + c ) (b+c) if b , c b,c are real numbers?


The answer is 7.

Yash Jain by Yash Jain , 21, India

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1 solution

Tom Engelsman
Mar 29, 2017

If one root is complex, then x 2 + b x + c = 0 x^2 + bx + c = 0 has a complex conjugate pair of roots (namely, 3 ± 2 i 3 \pm 2i .) We then obtain:

x 2 + b x + c = [ x − ( 3 + 2 i ) ] [ x − ( 3 − 2 i ) ] = x 2 − 6 x + 13 = 0 x^2 + bx + c = [x - (3+2i)][x - (3-2i)] = x^2 - 6x + 13 = 0

or, b + c = − 6 + 13 = 7 . b + c = -6 + 13 = \boxed{7}.

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