Multiple choice

If $\alpha,\beta$ are the roots of the equation $x^{2}+6x+9=0$, then the equation whose roots are $\alpha-3,\beta-3$ is

  1. $x^{2}+12x+36=0$
  2. $x^{2}-12x+36=0$
  3. $x^{2}-8x-36=0$
  4. $x^{2}-4x+18=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The roots of x^2 + 6x + 9 = 0 are -3 and -3. The new roots are alpha - 3 = -6 and beta - 3 = -6. The quadratic equation with roots -6 and -6 is (x + 6)(x + 6) = x^2 + 12x + 36 = 0.

AI explanation

Since alpha and beta are the roots of the equation x^2 + 6x + 9 = 0, their sum is alpha + beta = -6 and their product is alpha*beta = 9. For the new roots alpha - 3 and beta - 3, the sum becomes (alpha + beta) - 6 = -6 - 6 = -12, and the product becomes alpha*beta - 3(alpha + beta) + 9 = 9 - 3(-6) + 9 = 36. Using these new roots, the required quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0, which is x^2 + 12x + 36 = 0.

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