Question

How do you write a quadratic equation with a root of `-3+2i`?

Answer 1

`x^2 + 6x + 13`

Explanation:

If one root is `-3 + 2i`, then another root must be `-3 - 2i`. (When you solve the quadratic equation, there is a `+-` in front of the square root, so roots always come in pairs.)

We can use the sum and product of the roots to create a quadratic equation.

1. Find the sum of the roots:

`(-3 + 2i) + (-3 - 2i)`
`= -3 + 2i - 3 - 2i`
`= -3 + cancel(2i) - 3 - cancel(2i)`
`= -6`

2. Find the product of the roots:

`(-3 + 2i) * (-3 - 2i)`
`= 9 + 6i - 6i -4i^2`
`= 9 + cancel(6i) - cancel(6i) -4(-1)`
`= 13`

3. Use the formula `x^2 - Sx + P` and plug in the sum for S and the product for P.

`x^2 - (-6)x + 13`
`x^2 + 6x + 13`

This is your quadratic equation!