Question
What is the quadratic equation that has the roots 3+i and 3−i ?(1 point)
Responses
x2+6x−10=0
x squared plus 6 x minus 10 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+8=0
Responses
x2+6x−10=0
x squared plus 6 x minus 10 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+8=0
Answers
GPT-4o mini
To find the quadratic equation with roots \(3 + i\) and \(3 - i\), we can use the fact that if \(r_1\) and \(r_2\) are the roots, the quadratic equation can be expressed as:
\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0
\]
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:
\[
r_1 \cdot r_2 = (3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 10
\]
3. Substitute the values into the quadratic formula:
\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = x^2 - 6x + 10 = 0
\]
Thus, the quadratic equation is:
\[
x^2 - 6x + 10 = 0
\]
The correct response is:
**x²−6x+10=0**
\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0
\]
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:
\[
r_1 \cdot r_2 = (3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 10
\]
3. Substitute the values into the quadratic formula:
\[
x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = x^2 - 6x + 10 = 0
\]
Thus, the quadratic equation is:
\[
x^2 - 6x + 10 = 0
\]
The correct response is:
**x²−6x+10=0**