Question

Which values of x and y would make the following expression represent a real number? (4+5i)(x+yi)

Answer 1

Answer:

So x=4 and y=-5 would work.

Step-by-step explanation:

When you multiply complex conjugates, you get a real result.

Example:

$(a+bi)(a-bi)=a^2-b^2i^2=a^2+b^2$

I replace $i^2$ with -1 since $i^2=-1$.

$a^2+b^2$ is a real number (there is no imaginary part, no i).

So what is the conjugate of 4+5i?

4-5i

So x=4 and y=-5 would work.

A multiple of the complex conjugate would work as well.

$(a+bi)(ca-cbi)$

$c(a+bi)(a-bi)$

$c(a^2-b^2i^2)$

$c(a^2+b^2)$

This is still a real number; there is no imaginary part,no i.