Multiple choice

The equation $\displaystyle x^{2} +ax+b^{2 }=0$ has two roots each of which exceeds a number $c$, then

  1. $\displaystyle a^{2} < 4b^{2}$
  2. $\displaystyle c^{2} +ac+b^{2 }> 0$
  3. $\displaystyle -\dfrac{a}{2}< c$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a quadratic f(x) = x^2 + ax + b^2 to have roots > c, the vertex -a/2 must be > c, f(c) > 0, and the discriminant D = a^2 - 4b^2 >= 0. Option B, c^2 + ac + b^2 > 0, is equivalent to f(c) > 0, which is a necessary condition for the roots to be greater than c.

AI explanation

If both roots of the equation exceed c, then evaluating the quadratic polynomial f(x) = x^2 + ax + b^2 at x = c must yield a positive result because the parabola opens upwards and c lies to the left of both roots. This condition is written as f(c) > 0. Substituting x = c gives the necessary condition c^2 + ac + b^2 > 0.

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