Given the root [tex]\(3 + 2i\)[/tex] of a quadratic function, we need to complete the following statements and determine the quadratic function.
1. The other root, the conjugate, is [tex]\(\square\)[/tex]:
When dealing with polynomials with real coefficients, if a complex number [tex]\(a + bi\)[/tex] is a root, then its complex conjugate [tex]\(a - bi\)[/tex] must also be a root.
Therefore, the other root, the conjugate, is [tex]\(3 - 2i\)[/tex].
2. The quadratic function [tex]\(y\)[/tex] is given by:
For a quadratic equation with roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex], the equation is:
[tex]\[
y = x^2 - (\alpha + \beta)x + \alpha\beta
\][/tex]
Here, [tex]\(\alpha\)[/tex] is [tex]\(3 + 2i\)[/tex] and [tex]\(\beta\)[/tex] is [tex]\(3 - 2i\)[/tex].
- Sum of the roots [tex]\((\alpha + \beta)\)[/tex]:
[tex]\[
(3 + 2i) + (3 - 2i) = 3 + 3 + 2i - 2i = 6
\][/tex]
- Product of the roots [tex]\((\alpha \beta)\)[/tex]:
[tex]\[
(3 + 2i)(3 - 2i) = 3^2 - (2i)^2 = 9 - (4(-1)) = 9 + 4 = 13
\][/tex]
Therefore, the quadratic function is:
[tex]\[
y = x^2 - 6x + 13
\][/tex]
To summarize, the statements can be completed as follows:
The other root, the conjugate, is [tex]\(3 - 2i\)[/tex].
The quadratic function is:
[tex]\[
y = x^2 - 6x + 13
\][/tex]
