Multiple choice

Form the quadratic equation with rational coefficients whose one of the root is: $2\, +\, 3\sqrt5$

  1. $x^{2}\, -\, 4x\, -\, 41\,  =\, 0$
  2. $x^{2}\, +\, 4x\, -\, 41\,  =\, 0$
  3. $x^{2}\, -\, 4x\, -\, 45\,  =\, 0$
  4. $x^{2}\, +\, 4x\, -\, 45\,  =\, 0$
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A Correct answer
Explanation

If one root is 2 + 3*sqrt(5), the other must be 2 - 3*sqrt(5) for rational coefficients. Sum of roots = 4. Product of roots = 2^2 - (3*sqrt(5))^2 = 4 - 45 = -41. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 4x - 41 = 0.

AI explanation

For rational coefficients, the conjugate of the given root, which is 2 - 3√5, must also be a root. The sum of the roots is (2 + 3√5) + (2 - 3√5) = 4, and the product is (2 + 3√5)(2 - 3√5) = 2^2 - (3√5)^2 = 4 - 45 = -41. Using the formula x^2 - (sum)x + (product) = 0, the equation is x^2 - 4x - 41 = 0.

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