To find the quadratic equation with roots \(3+i\) and \(3-i\), we can use Vieta's formulas. The sum of the roots (\(r_1 + r_2\)) and the product of the roots (\(r_1 \cdot r_2\)) can be used to form the quadratic equation in the standard form \(x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0\).
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Sum of the roots: \[ (3+i) + (3-i) = 3 + i + 3 - i = 6 \]
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Product of the roots: \[ (3+i)(3-i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]
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Form the quadratic equation: Using the sum and product derived above, we can write the equation: \[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \] Plugging in the values we found, we get: \[ x^2 - 6x + 10 = 0 \]
So, the correct quadratic equation is:
\[ x^2 - 6x + 10 = 0 \]
Thus, the answer is \(x^2 - 6x + 10 = 0\).