Chapter 7.7, Problem 20WE

To determine

Tofind: thequadratic equation.

Answer to Problem 20WE

The quadratic equation is $x^2-8x+20=0$ .

Explanation of Solution

Given information:

The roots are $4+2i,4-2i$ .

Calculation: to find the quadratic equation, first find the sum of roots and products of roots,

$\begin{array}{l}\text{sum of roots}=4-2i+4+2i\\ \text{sum of roots}=8\text{ [add]}\\ \text{product of roots}=\left(4-2i\right)\left(4+2i\right)\\ \text{product of roots}=16-4i^2\text{ [multiply]}\\ \text{product of roots}=20\text{ [}{i}^2=-1]\end{array}$

Now, substitute in the formula and simplify,

$\begin{array}{l}{x}^2-\left(\text{sum of roots}\right)x+\left(\text{product of roots}\right)=0\\ x^2-\left(8\right)x+\left(20\right)=0\text{ [substitute]}\\ x^2-8x+20=0\text{ [simplify]}\end{array}$

Thus, the quadratic equation is $x^2-8x+20=0$ .