Multiple choice

The sum of reciprocals of the roots of the equation $ax^2+bx+c=0$, $a, b,$ $c\ne0$, is _______.

  1. $\displaystyle\frac{1}{a}+\frac{1}{b}$
  2. $\displaystyle\frac{1}{b}+\frac{1}{c}$
  3. $\displaystyle\frac{c}{b}$
  4. $\displaystyle-\frac{b}{c}$
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D Correct answer
Explanation

If the roots are p and q, then p+q = -b/a and pq = c/a. The sum of reciprocals is (1/p) + (1/q) = (p+q)/(pq) = (-b/a) / (c/a) = -b/c.

AI explanation

If the roots of the quadratic equation ax^2 + bx + c = 0 are alpha and beta, their sum is -b/a and their product is c/a. The sum of their reciprocals is 1/alpha + 1/beta, which combines to (alpha + beta) / (alpha)(beta). Substituting the known sum and product into this expression gives (-b/a) / (c/a). Simplifying the fraction yields -b/c as the sum of the reciprocals of the roots.

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