Multiple choice

The quadratic function $f(x)$ which have one roots as $\displaystyle 2+\sqrt{3}$, is

  1. $x^2-4x+1$
  2. $x^2+4x-1$
  3. $x^2-4x-1$
  4. $x^2+2x-1$
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A Correct answer
Explanation

If one root is 2 + sqrt(3), the other must be 2 - sqrt(3). The sum of roots is 4 and the product is (2+sqrt(3))(2-sqrt(3)) = 4-3 = 1. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 4x + 1 = 0.

AI explanation

For a quadratic equation with rational coefficients, irrational roots occur in conjugate pairs, so the other root must be 2 - sqrt(3). Using the sum of roots (2 + sqrt(3) + 2 - sqrt(3) = 4) and product of roots ((2 + sqrt(3))*(2 - sqrt(3)) = 4 - 3 = 1), we can form the equation. The quadratic function is x^2 - (sum)x + (product) = 0, which is x^2 - 4x + 1.

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