Question

Find two complex numbers that have a sum of 10i, a difference of -4 and a product of -29

Answer 1

Let these two numbers be $z$ and $w$. Their sum is $10i$, their difference is -4, and their product is -29:

$z+w=10i$

$z-w=-4$

$zw=-29$

Add the first two equations together to eliminate $w$:

$(z+w)+(z-w)=10i-4\implies 2z=-4+10i\implies z=-2+5i$

Then

$w=10i-z\implies w=10i-(-2+5i)=2+5i$

Just to confirm this is correct, check their product:

$(-2+5i)(2+5i)=-4+25i^2=-4-25=-29$