Multiple choice

If r be the ratio of the roots of the equation $ax^2+bx+c=0$, then $\dfrac{(r+1)^2}{r}=$

  1. $\dfrac{a^2}{bc}$
  2. $\dfrac{b^2}{ca}$
  3. $\dfrac{c^2}{ab}$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let roots be alpha and beta. r = alpha/beta. alpha + beta = -b/a, alpha * beta = c/a. (r+1)^2 / r = (alpha/beta + 1)^2 / (alpha/beta) = ((alpha+beta)/beta)^2 * (beta/alpha) = (alpha+beta)^2 / (alpha*beta) = (-b/a)^2 / (c/a) = b^2 / ac.

AI explanation

Let the roots be alpha and r * alpha, where r is the given ratio. The sum of the roots is alpha(r + 1) = -b/a, and the product of the roots is r * alpha^2 = c/a. To find the value of (r + 1)^2 / r, we square the sum equation and divide it by the product equation: (alpha^2 * (r + 1)^2) / (r * alpha^2) = (b^2 / a^2) / (c/a). This simplifies directly to b^2 / (ac).

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