Multiple choice

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ find the equation having roots $\dfrac{1}{\alpha}$ and $\dfrac{1}{\beta}$.

  1. $bx^2 + cx + a = 0$
  2. $cx^2 + ax + b = 0$
  3. $cx^2 + bx + a = 0$
  4. $cx^2 - bx + a = 0$
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C Correct answer
Explanation

If the roots are alpha and beta, the equation is a(x-alpha)(x-beta) = ax^2 - a(alpha+beta)x + a(alpha*beta) = 0. Comparing to ax^2 + bx + c = 0, we have alpha+beta = -b/a and alpha*beta = c/a. For roots 1/alpha and 1/beta, the new sum is (alpha+beta)/(alpha*beta) = (-b/a)/(c/a) = -b/c, and the product is 1/(alpha*beta) = a/c. The equation is x^2 - (sum)x + (product) = 0, which is x^2 + (b/c)x + (a/c) = 0, or cx^2 + bx + a = 0.

AI explanation

If alpha and beta are the roots of ax^2 + bx + c = 0, replacing every x with 1/x gives the equation a/x^2 + b/x + c = 0. Multiplying the entire equation by x^2 to clear the denominators results in cx^2 + bx + a = 0. This new equation has roots 1/alpha and 1/beta.

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