To find the quadratic equation with roots [tex]\(x = 2 + i\)[/tex] and [tex]\(x = 2 - i\)[/tex], follow these steps:
1. Identify the Roots:
The given roots are [tex]\( x = 2 + i \)[/tex] and [tex]\( x = 2 - i \)[/tex].
2. Form the Factors of the Roots:
The quadratic equation with these roots can be formed by setting up the factors:
[tex]\[
(x - (2 + i))(x - (2 - i)) = 0
\][/tex]
3. Simplify the Expression:
Rewrite the expression to distribute and simplify it step-by-step:
[tex]\[
(x - 2 - i)(x - 2 + i)
\][/tex]
This can be seen as a product of a sum and difference of terms:
[tex]\[
[(x - 2) - i][(x - 2) + i]
\][/tex]
4. Apply the Difference of Squares Formula:
Using the formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex] where [tex]\(a = x - 2\)[/tex] and [tex]\(b = i\)[/tex], we get:
[tex]\[
(x - 2)^2 - (i)^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
(x - 2)^2 - (-1)
\][/tex]
Simplifying this gives:
[tex]\[
(x - 2)^2 + 1
\][/tex]
5. Expand [tex]\((x - 2)^2\)[/tex]:
Expand the binomial:
[tex]\[
(x - 2)^2 = x^2 - 4x + 4
\][/tex]
6. Form the Quadratic Equation:
Substitute back into the equation:
[tex]\[
(x - 2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5
\][/tex]
Thus, the quadratic equation with the roots [tex]\(x = 2 + i\)[/tex] and [tex]\(x = 2 - i\)[/tex] is:
[tex]\[
x^2 - 4x + 5 = 0
\][/tex]
So, the correct quadratic equation is:
[tex]\(\boxed{x^2 - 4x + 5 = 0}\)[/tex]
