Determine all real values of A for which there exist distinct complex numbers x1,x2 such that the following three equations hold: x1(x1+1)=A x_{2}(x_{2}+1) =A x14+3x13+5x1=x24+3x23+5x2
A number or a short expression. Spacing and $ signs are ignored.
Solution
Applying polynomial division, x14+3x13+5x1=(x12+x1−A)(x12+2x1+(A−2))+(A+7)x1+A(A−2)=(A+7)x1+A(A−2). Thus, in order for the last equation to hold, we need (A+7)x1=(A+7)x2, from which it follows that A=−7. These steps are reversible, so A=−7 indeed satisfies the needed condition.
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