# Complex numbers question ;)

## Homework Statement

One root of the equation x^2 + ax + b = 0 is 4 + 5i.

Write down the second root.

N/a?

## The Attempt at a Solution

My problem is it's a "write down" question which suggests no working required. This is probably so simple but I just don't know... I cannot do this even thought I know the later parts of the question. Thank you in advance ;)

## Answers and Replies

Pengwuino
Gold Member
Write down the quadratic equation as you know it. The quadratic formula gives you two equations using one formula. What changes about the formula that gives you 2 separate solutions?

The plus or minus, so would it be 4 - 5i?

Pengwuino
Gold Member
Yup! The quadratic formula gives solutions as

$$$\frac{{ - b}}{{2a}} \pm \frac{{\sqrt {b^2 - 4ac} }}{{2a}}$$$

However, you can rewrite the quadratic formula by pulling out a -1 from the term in the square root to get

$$\[ \frac{{ - b}}{{2a}} \pm i\frac{{\sqrt {4ac - b^2 } }}{{2a}}$$

So you can identify the 4 and the 5 with the real and imaginary parts of the equation.

Hehe, thank you so much :)

I know it was rather simple but my mind just wouldn't trigger.

Ty again ;)

jbunniii
Science Advisor
Homework Helper
Gold Member
By the way, that answer is correct only if $a$ and $b$ are both real. If the problem doesn't specify that they are, then the other root could be any complex number.