Multiple choice

If the equation $ ax^2 + bx + c = 0, a > 0 $ has two distinct real roots $ \alpha $ and $ \beta $ such that $ \alpha < - 5 $ and $ \beta > 5, $ then :

  1. $ c > 0 $
  2. $ c = 0 $
  3. $ c = \dfrac {a+b}{2} $
  4. $ c < 0 $
  5. $ c = a +b $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a quadratic f(x) = ax^2 + bx + c with a > 0, if the roots are on opposite sides of 0 (specifically alpha < -5 and beta > 5), then f(0) must be negative. Since f(0) = c, c must be less than 0.

AI explanation

The condition states the quadratic equation ax^2 + bx + c = 0 has roots alpha < -5 and beta > 5. This means the graph of y = ax^2 + bx + c crosses the x-axis once to the left of -5 and once to the right of 5. Therefore, the value of the quadratic function at x = 0 must lie below the x-axis, so f(0) < 0. Substituting x = 0 gives c < 0.

Keep practicing — related questions