Question

What is the quadratic equation that has the roots  3+i  and  3−i ?(1 point)Responsesx2−6x+8=0x squared minus 6 x plus 8 equals 0x2+6x+10=0x squared plus 6 x plus 10 equals 0x2−6x+10=0x squared minus 6 x plus 10 equals 0x2+6x−10=0

Answer 1

To find the quadratic equation with the roots \(3+i\) and \(3-i\), we can use the fact that if a quadratic equation has roots \(r_1\) and \(r_2\), it can be expressed as:

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

In this case, the roots are \(r_1 = 3+i\) and \(r_2 = 3-i\).

1. **Sum of the roots**:
\[
r_1 + r_2 = (3+i) + (3-i) = 3 + 3 + i - i = 6
\]

2. **Product of the roots**:
\[
r_1 \cdot r_2 = (3+i)(3-i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10
\]

Now we can substitute these values back into the quadratic equation format:

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

Thus, the quadratic equation is:

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

So the correct response is:

**x² - 6x + 10 = 0**.