Find two complex numbers that have a sum of 10i, a difference of -4 and a product of -29
Accepted Solution
A:
Let these two numbers be [tex]z[/tex] and [tex]w[/tex]. Their sum is [tex]10i[/tex], their difference is -4, and their product is -29:[tex]z+w=10i[/tex][tex]z-w=-4[/tex][tex]zw=-29[/tex]Add the first two equations together to eliminate [tex]w[/tex]:[tex](z+w)+(z-w)=10i-4\implies 2z=-4+10i\implies z=-2+5i[/tex]Then[tex]w=10i-z\implies w=10i-(-2+5i)=2+5i[/tex]Just to confirm this is correct, check their product:[tex](-2+5i)(2+5i)=-4+25i^2=-4-25=-29[/tex]