# How do you find the quadratic equation that has the following roots: 4+-3isqrt2?

Sep 12, 2017

${x}^{2} - 8 x + 34 = 0$

#### Explanation:

We are given that the quadratic sought has roots $4 \pm 3 i \sqrt{2}$, denote the roots by:

$\alpha = 4 - 3 i \sqrt{2}$
$\beta = 4 + 3 i \sqrt{2}$

So, given these roots, we have:

$\alpha + \beta = 4 - 3 i \sqrt{2} + 4 + 3 i \sqrt{2}$
$\text{ } = 8$

$\alpha \setminus \beta = \left(4 - 3 i \sqrt{2}\right) \left(4 + 3 i \sqrt{2}\right)$
$\text{ } = {\left(4\right)}^{2} - {\left(3 i \sqrt{2}\right)}^{2}$
$\text{ } = 16 + 9 \cdot 2$
$\text{ } = 16 + 18$
$\text{ } = 34$

And so the equation we seek ijus:

${x}^{2} - \left(\alpha + \beta\right) x + \alpha \beta = 0$
$\therefore {x}^{2} - 8 x + 34 = 0$