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 of a quadratic equation in the form \(x^2 + bx + c = 0\), then:
- The sum of the roots \(r_1 + r_2 = -b\)
- The product of the roots \(r_1 \times r_2 = c\)
In this case:
- \(r_1 = 3 + i\)
- \(r_2 = 3 - i\)
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Calculate the sum of the roots: \[ r_1 + r_2 = (3 + i) + (3 - i) = 3 + i + 3 - i = 6 \]
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Calculate the product of the roots: \[ r_1 \times r_2 = (3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]
Now we will use the relationships:
- The sum of the roots gives us \( -b = -6 \) (thus \( b = -6 \))
- The product of the roots gives us \( c = 10 \)
Therefore, the quadratic equation is: \[ x^2 - 6x + 10 = 0 \]
The correct response is: x²−6x+10=0 (x squared minus 6 x plus 10 equals 0)