Question

The roots of the quadratic equation $z^2 + az + b = 0$ are $2 - 3i$ and $2 + 3i$. What is $a+b$?

Answer 1

`a + b = 9`

Explanation:

We have a quadratic equation in z, given as:

`{z}^(2) + az + b = 0`

with roots

`z_1 = 2 + 3i \: and \: z_2 = 2 - 3i`

The sum of roots is given by:

`z_1 + z_2 = - (a)/(1)`

This implies that

`2 - 3i + 2 + 3i= - (a)/(1)`

`4 = - a`

`a = - 4`

Also product of roots is given by:

`z_1z_2 = (b)/(1)`

This implies that:

`(2 + 3i)(2 - 3i) = b`

`b = {2}^(2) + {3}^(2)`

`b = 13`

Therefore

`a + b = 13 - 4 = 9`