Multiple choice

The equation whose roots are $\dfrac{1}{3 + \sqrt 2} \ and \ \dfrac{1}{3 - \sqrt 2}$ is

  1. $7x^2 - 6x + 1 =0$
  2. $6x^2 - 6x + 1 =0$
  3. $x^2 - 6x + 7 =0$
  4. $x^2 - 7x + 6 =0$
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AI explanation

To form a quadratic equation from its roots, we use the sum of the roots (S) and the product of the roots (P) in the formula x^2 - Sx + P = 0. The sum S is calculated as (1 / (3 + sqrt(2)) + 1 / (3 - sqrt(2))), which combines to (6 / 7) after rationalization. The product P is 1 / ((3 + sqrt(2))(3 - sqrt(2))), which equals 1 / 7. Substituting these values gives x^2 - (6/7)x + 1/7 = 0, which simplifies by multiplying by 7 to 7x^2 - 6x + 1 = 0.

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