Complex Quadratic

Algebra Level 3

x 2 + 3 i x + 4 = 0 \large{x^{2} + 3ix+4 = 0}

Suppose the roots of the above equation are A A and B B , enter the value of A 2 + B 2 A^{2}+B^{2} .

Clarification : i = − 1 i=\sqrt{-1} denotes the imaginary unit .


The answer is -17.

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Steven Chase by Steven Chase , 37, USA

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2 solutions

By Vieta's A + B = − 3 i A + B = -3i and A B = 4 AB = 4 .

Thus A 2 + B 2 = ( A + B ) 2 − 2 A B = ( − 3 i ) 2 − 2 × 4 = − 9 − 8 = − 17 A^{2} + B^{2} = (A + B)^{2} - 2AB = (-3i)^{2} - 2 \times 4 = -9 - 8 = \boxed{-17} .

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Steven Chase
Sep 21, 2016

The good ol' quadratic formula works too.

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