Multiple choice

If $\alpha, \beta$ are root of the equation $x^2 -5x+6=0$ then the equation whose roots are $\alpha +3$ and $\beta +3$ is

  1. $x^2 -11x+30=0$
  2. $(x-3)^2 - 5 (x-3)+6=0$

  3. Both (A) and (B)

  4. None
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If roots are alpha and beta, the new roots are alpha+3 and beta+3. Let y = x+3, so x = y-3. Substituting into the original equation: (y-3)^2 - 5(y-3) + 6 = 0. This matches option B. Expanding this gives y^2 - 6y + 9 - 5y + 15 + 6 = y^2 - 11y + 30 = 0, which matches option A. Thus, both are correct.

AI explanation

To find the equation whose roots are increased by 3, substitute (x minus 3) for x in the original equation. This yields (x minus 3) squared minus 5 times (x minus 3) plus 6 equals 0. Expanding this gives x squared minus 6x plus 9 minus 5x plus 15 plus 6 equals 0, which simplifies to x squared minus 11x plus 30 equals 0. Both forms are correct.

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