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

Jun 21, 2017

${x}^{2} + 6 x + 13$

#### Explanation:

If one root is $- 3 + 2 i$, then another root must be $- 3 - 2 i$. (When you solve the quadratic equation, there is a $\pm$ 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:

$\left(- 3 + 2 i\right) + \left(- 3 - 2 i\right)$
$= - 3 + 2 i - 3 - 2 i$
$= - 3 + \cancel{2 i} - 3 - \cancel{2 i}$
$= - 6$

2. Find the product of the roots:

$\left(- 3 + 2 i\right) \cdot \left(- 3 - 2 i\right)$
$= 9 + 6 i - 6 i - 4 {i}^{2}$
$= 9 + \cancel{6 i} - \cancel{6 i} - 4 \left(- 1\right)$
$= 13$

3. Use the formula ${x}^{2} - S x + P$ and plug in the sum for S and the product for P.

${x}^{2} - \left(- 6\right) x + 13$
${x}^{2} + 6 x + 13$