Multiple choice

If x1 and x2 are the roots of the equation x2 - 2x + 4 = 0, what is the equation with roots 3x1 - 2 and 3x2 - 2?

  1. x2 - 2x - 28 = 0

  2. x2 + 2x + 28 = 0

  3. x2 + 2x - 28 = 0

  4. x2 - 2x + 28 = 0

  5. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Roots x1, x2 satisfy x^2 - 2x + 4 = 0. x1+x2 = 2, x1x2 = 4. New roots y1 = 3x1-2, y2 = 3x2-2. Sum = 3(x1+x2) - 4 = 3(2) - 4 = 2. Product = (3x1-2)(3x2-2) = 9x1x2 - 6(x1+x2) + 4 = 9(4) - 6(2) + 4 = 36 - 12 + 4 = 28. Equation: x^2 - (sum)x + product = 0 => x^2 - 2x + 28 = 0.

AI explanation

From the given equation x2 - 2x + 4 = 0, the sum of the roots is x1 + x2 = 2 and the product of the roots is x1 x2 = 4. For the new roots y1 = 3x1 - 2 and y2 = 3x2 - 2, the sum is y1 + y2 = 3(x1 + x2) - 4 = 3(2) - 4 = 2, and the product is (3x1 - 2)(3x2 - 2) = 9(x1 x2) - 6(x1 + x2) + 4 = 9(4) - 6(2) + 4 = 28. The required quadratic equation is x2 - (sum of roots)x + (product of roots) = 0, which becomes x2 - 2x + 28 = 0.

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