Multiple choice

If $a, b, c \epsilon R, a>0 and c \neq 0$let $\alpha and \beta $ be the real and distinct roots of the equation $ ax^2+bx+c=\left | c \right | $ and p,q be the real and distinct roots of the equation $ax^2+bx+c=0$. Then

  1. p and q lie between $ \alpha $ and $\beta$
  2. p and q do not lie between $ \alpha $ and $\beta$
  3. only p lie between $ \alpha $ and $\beta$
  4. only q lie between $ \alpha $ and $\beta$
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A Correct answer
Explanation

The roots of ax^2+bx+c=0 are p, q. The roots of ax^2+bx+c=|c| are alpha, beta. Since |c| > 0, the parabola ax^2+bx+c is shifted downwards by |c| to reach zero, meaning the roots of the first equation are 'inside' the roots of the second.

AI explanation

Let f(x) = ax^2 + bx + c and note that the parabola opens upward because a is greater than 0. The roots alpha and beta belong to the equation ax^2 + bx + c - |c| = 0, while p and q are the roots of f(x) = ax^2 + bx + c = 0. Because |c| is strictly greater than c for any negative c and strictly greater than zero, analyzing the vertical shift of the parabola shows that the roots of f(x) = |c| are always closer to the vertex than the roots of f(x) = 0. Therefore, the interval (p, q) is completely contained within the interval (alpha, beta). Consequently, both p and q lie strictly between alpha and beta.

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