Multiple choice

What is the condition that the equation $ax^{2} + bx + c = 0$, where $a\neq 0$, has both the roots positive?

  1. $a, b $ and $c$ are of same sign
  2. $a$ and $b$ are of same sign
  3. $b$ and $c$ have the same sign opposite to that of $a$
  4. $a$ and $c$ have the same sign opposite to that of $b$
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D Correct answer
Explanation

For roots to be positive, the product of roots (c/a) must be positive, and the sum of roots (-b/a) must be positive. This implies a and c have the same sign, and a and b have opposite signs.

AI explanation

For both roots to be positive, the product of the roots must be positive and their sum must be positive. Since the product of the roots equals c divided by a, the coefficients a and c must share the same sign. Because the sum of the roots equals negative b divided by a, the coefficient b must have the opposite sign of a. Therefore, a and c share the same sign, which is opposite to that of b.

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