Question

For what value of "a" will the product shown below have a purely imaginary value? (Show work please)

(4+5i)(a+2i)

Answer 1

If you expand out the brackets you get this,

(4+5i)(a+2i) = 4a + (5a)i + 8i - 10

The -10 comes from 5i * 2i.
Squaring i becomes -1.

Let's group the real stuff together,
and imaginary separately,

(4a - 10) + (5a + 8)i

For this to be purely imaginary,
the real part needs to be zero.

Therefore 4a - 10 = 0

Solve for a.