Multiple choice

If $p$ and $q$ are the roots of the equation $ax^2 +bx +c =0$, then the value of $\dfrac {p}{q}+\dfrac {q}{p}$ is

  1. $b^2-2ac$
  2. $\dfrac {b^2-2ac}{ac}$
  3. $\dfrac {b^2-4ac}{2a}$
  4. $\dfrac {b^2-4ac}{ac}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For ax^2 + bx + c = 0, the sum of roots p + q = -b/a and product pq = c/a. The expression p/q + q/p = (p^2 + q^2) / pq = ((p + q)^2 - 2pq) / pq. Substituting the values: ((-b/a)^2 - 2(c/a)) / (c/a) = (b^2/a^2 - 2c/a) / (c/a) = (b^2 - 2ac) / ac.

AI explanation

For the quadratic equation ax^2 + bx + c = 0, the sum of roots p and q is -b/a and the product of roots is c/a. The expression p/q + q/p can be combined as (p^2 + q^2)/(pq). Using the algebraic identity p^2 + q^2 = (p + q)^2 - 2pq, substituting the sum and product values gives ((-b/a)^2 - 2(c/a))/(c/a). Simplifying the numerator results in (b^2/a^2 - 2c/a), which equals (b^2 - 2ac)/a^2; dividing this by the product c/a yields (b^2 - 2ac)/(ac).

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