Complex roots

Algebra Level 2

If one of the roots of the equation:- a x 2 + b x + c = 0 ax^{2}+bx+c=0 is 1 − i 1-i and gcd(a,b,c)=1 calculate the value of a+b+c


The answer is 1.

Aman Sharma by Aman Sharma , 25, India

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1 solution

Tom Engelsman
Feb 15, 2017

If one root is complex, then the other root will its complex conjugate (or 1 + i . 1 + i. ) The above quadratic can then be expressed as:

[ x − ( 1 − i ) ] [ x − ( 1 + i ) ] = 0 ⇒ x 2 − [ ( 1 − i ) + ( 1 + i ) ] x + ( 1 − i 2 ) = 0 ⇒ x 2 − 2 x + 2 = 0 [x - (1-i)][x - (1+i)] = 0 \Rightarrow x^2 - [(1-i)+(1+i)]x + (1 - i^{2}) = 0 \Rightarrow x^2 - 2x + 2 = 0

Thus, a = 1 , b = − 2 , c = 2 ⇒ a + b = c = 1 . a = 1, b = -2, c = 2 \Rightarrow a + b = c = \boxed{1}.

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