Finding the Second Root of a Complex Number Equation

[Maatttt0]

Homework Statement

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

Write down the second root.

Homework Equations

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 ;)

 

[Pengwuino]

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?

 

[Maatttt0]

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

 

[Pengwuino]

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.

 

[Maatttt0]

Hehe, thank you so much :)

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

Ty again ;)

 

[jbunniii]

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.