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 minus 10 equals 0

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

x2+6x+10=0

Answer 1

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

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

First, we calculate the sum and product of the roots:

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

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

Now, substitute the sum and product into the quadratic equation form:

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

which simplifies to:

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

Thus, the correct quadratic equation is:

\[
\boxed{x^2 - 6x + 10 = 0}.
\]