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

1 Answer
Jun 21, 2017

#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!