Multiple choice

Form a quadratic equations, whose roots are $2+\sqrt{2}$ and $2-\sqrt{2}$

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

If roots are r1 and r2, the equation is x^2 - (r1+r2)x + (r1*r2) = 0. Sum = (2+sqrt(2)) + (2-sqrt(2)) = 4. Product = (2+sqrt(2))(2-sqrt(2)) = 4 - 2 = 2. Equation: x^2 - 4x + 2 = 0.

AI explanation

Using the formula x^2 - (sum of roots)x + (product of roots) = 0, we calculate the sum of the roots as (2 + sqrt(2)) + (2 - sqrt(2)) = 4. The product of the roots is (2 + sqrt(2))(2 - sqrt(2)) = 4 - 2 = 2. Substituting these values gives the quadratic equation x^2 - 4x + 2 = 0.

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