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

f (x) = x^{2} - 4 x - 5

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f (x) = x^{2} - 4 x + 13

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f (x) = x^{2} - 6 i x - 5

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f (x) = x^{2} - 6 i x + 13

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Solution

The correct option is A $f (x) = x^{2} - 4 x + 13$
Let the equation $f (x)$ be $x^{2} + b x + c$
Sum of roots is equal to $- b = 2 - 3 i + 2 + 3 i = 4$ , which implies $b = - 4$
Product of roots is equal to $c = (2 - 3 i) (2 + 3 i) = 4 + 9 = 13$
therefore the equation is $x^{2} - 4 x + 13$

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