Question

What is the quadratic equation that has the roots 3+i and 3−i ?(1 point)
Responses

x2−6x+8=0
x squared minus 6 x plus 8 equals 0

x2+6x+10=0
x squared plus 6 x plus 10 equals 0

x2−6x+10=0
x squared minus 6 x plus 10 equals 0

x2+6x−10=0
x squared plus 6 x minus 10 equals 0

Answer 1

To find the quadratic equation with roots \( 3+i \) and \( 3-i \), we can use the fact that the quadratic can be expressed in the form:

\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0
\]

where \( r_1 \) and \( r_2 \) are the roots.

1. **Calculate the sum of the roots \( r_1 + r_2 \):**

\[
(3+i) + (3-i) = 3 + i + 3 - i = 6
\]

2. **Calculate the product of the roots \( r_1 \cdot r_2 \):**

\[
(3+i)(3-i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10
\]

3. **Substituting these values into the quadratic equation:**

\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = x^2 - 6x + 10 = 0
\]

Thus, the quadratic equation is:

\[
x^2 - 6x + 10 = 0
\]

So, the correct response is:

**x² - 6x + 10 = 0**