Multiple choice

Suppose that the function $f(x)$ is a quadratic function with roots at $x=2-3i$ and $x=2+3i$. Find $f(x)$.

  1. $f(x)=x^2-4x-5$
  2. $f(x)=x^2-4x+13$
  3. $f(x)=x^2-6ix-5$
  4. $f(x)=x^2-6ix+13$
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B Correct answer
Explanation

For a quadratic with roots 2+3i and 2-3i, the sum of roots is 4 and the product is (2+3i)(2-3i) = 4 + 9 = 13. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 4x + 13 = 0.

AI explanation

Using the sum and product of roots formula for a quadratic function, the sum is (2 - 3i) plus (2 + 3i) = 4, and the product is (2 - 3i) times (2 + 3i) = 4 minus 9(i^2) = 13. A quadratic function can be written as f(x) = x^2 minus the sum of the roots times x plus the product of the roots. Substituting the calculated values gives f(x) = x^2 - 4x + 13.

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