Chapter 7.7, Problem 22WE

To determine

Tofind: thequadratic equation.

Answer to Problem 22WE

The quadratic equation is $x^2+4x+11=0$ .

Explanation of Solution

Given information:

The roots are $-2+i\sqrt{7},-2-i\sqrt{7}$ .

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

$\begin{array}{l}\text{sum of roots}=-2-i\sqrt{7}-2+i\sqrt{7}\\ \text{sum of roots}=-4\text{ [add]}\\ \text{product of roots}=\left(-2-i\sqrt{7}\right)\left(-2+i\sqrt{7}\right)\\ \text{product of roots}=4-7i^2\text{ [multiply]}\\ \text{product of roots}=11\text{ [}{i}^2=-1]\end{array}$

Now, substitute in the formula and simplify,

$\begin{array}{l}{x}^2-\left(\text{sum of roots}\right)x+\left(\text{product of roots}\right)=0\\ x^2-\left(-4\right)x+\left(11\right)=0\text{ [substitute]}\\ x^2+4x+11=0\text{ [simplify]}\end{array}$

Thus, the quadratic equation is $x^2+4x+11=0$ .