Multiple choice

If the zeroes of a quadratic polynomial $ax^{2}+bx+c$ are both negative, then which of the following is always correct

  1. $\dfrac{-b}{a}>0$
  2. $\dfrac{c}{a}=0$
  3. $a,b$ and $c$ all are zero
  4. $a,b$ and $c$ all have same sign
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If both roots are negative, their sum (-b/a) must be negative, so b/a > 0. Their product (c/a) must be positive. For both to be positive, a, b, and c must have signs such that b/a > 0 and c/a > 0, implying a, b, and c share the same sign.

AI explanation

For a quadratic polynomial ax squared plus bx plus c to have two negative roots, the sum of the roots must be negative, and the product must be positive. By Vieta's formulas, the sum of the roots equals negative b divided by a, so if the sum is negative, the expression negative b divided by a must be negative, meaning a and b have the same sign. The product of the roots equals c divided by a, so if the product is positive, c divided by a must be positive, meaning a and c have the same sign. Therefore, a, b, and c must all have the same sign.

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