Multiple choice

If $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$, then the value of $\dfrac{(a \alpha^2 + c)}{(a \alpha + b)} + \dfrac{(a \beta^2 + c)}{(a \beta + b)}$ is

  1. $\displaystyle \frac{b (b^2 - 2ac)}{4a}$
  2. $\displaystyle \frac{b^2 - 4ac}{2a}$
  3. $\displaystyle \frac{b (b^2 - 2ac)}{a^2 c}$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given ax^2+bx+c=0, roots alpha, beta. a*alpha^2+c = -b*alpha. The expression becomes (-b*alpha)/(a*alpha+b) + (-b*beta)/(a*beta+b). This simplifies to -b * (alpha/(a*alpha+b) + beta/(a*beta+b)). Using alpha+beta = -b/a and alpha*beta = c/a, the calculation leads to b(b^2-2ac)/(a^2*c).

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