Multiple choice

Form the quadratic equation with rational coefficients whose one of the root is: $3\, -\, 2\sqrt3$

  1. $x^{2}\, -\, 3x\, -\, 6\,  =\, 0$
  2. $x^{2}\, +\, 6x\, -\, 3\,  =\, 0$
  3. $x^{2}\, +\, 3x\, -\, 6\,  =\, 0$
  4. $x^{2}\, -\, 6x\, -\, 3\,  =\, 0$
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D Correct answer
Explanation

If one root is 3 - 2*sqrt(3), the other must be 3 + 2*sqrt(3) for rational coefficients. Sum of roots = 6, product = 9 - 12 = -3. Equation: x^2 - (sum)x + product = 0, which is x^2 - 6x - 3 = 0.

AI explanation

For rational coefficients, the conjugate of the given root, which is 3 + 2√3, must also be a root. The sum of these roots is (3 - 2√3) + (3 + 2√3) = 6, and the product is (3 - 2√3)(3 + 2√3) = 3^2 - (2√3)^2 = 9 - 12 = -3. Using the formula x^2 - (sum)x + (product) = 0, the equation is x^2 - 6x - 3 = 0.

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