Multiple choice

If $a,b$ are the roots of a equation of a quadratic equation $x^2-3x+5=0$ then the equation whose roots are $(a^2-3a+7)$ and $(b^2-3b+7)$ is-

  1. $x^2+4x+1=0$
  2. $x^2-4x-1=0$
  3. $x^2+4x+4=0$
  4. $x^2-4x+4=0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given a, b are roots of x^2 - 3x + 5 = 0, so a^2 - 3a + 5 = 0 and b^2 - 3b + 5 = 0. Thus, a^2 - 3a = -5 and b^2 - 3b = -5. The new roots are (-5 + 7) = 2 and (-5 + 7) = 2. The equation with roots 2, 2 is (x-2)^2 = x^2 - 4x + 4 = 0.

AI explanation

Because a is a root of the equation x^2 - 3x + 5 = 0, we know a^2 - 3a + 5 = 0, which can be rearranged to a^2 - 3a = -5. Adding 7 to both sides gives a^2 - 3a + 7 = 2, and the same logic applies to the second root b, meaning the new roots are both 2. A quadratic equation with roots 2 and 2 is formed using Vieta's formulas, where the sum of the roots is 4 and the product is 4. The required equation is x^2 - 4x + 4 = 0.

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