Multiple choice

If $a, b$ and $c$ are in arithmetic progression, then the roots of the equation $ax^{2} - 2bx + c = 0$ are

  1. $1$ and $\dfrac {c}{a}$
  2. $-\dfrac {1}{a}$ and $-c$
  3. $-1$ and $-\dfrac {c}{a}$
  4. $-2$ and $-\dfrac {c}{2a}$
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A Correct answer
Explanation

Since a, b, c are in A.P., 2b = a + c. Substituting this into the equation ax^2 - (a+c)x + c = 0, we get ax^2 - ax - cx + c = 0, which factors to ax(x-1) - c(x-1) = 0. Thus, (ax-c)(x-1) = 0, yielding roots 1 and c/a.

AI explanation

Since a, b, and c are in arithmetic progression, the middle term can be expressed as 2b = a + c. Substituting 2b into the quadratic equation yields ax^2 - (a + c)x + c = 0. Factoring this by grouping gives ax(x - 1) - c(x - 1) = 0, which leads to the factors (x - 1)(ax - c) = 0. Equating each factor to zero provides the roots as 1 and c/a.

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