Multiple choice

lf both the roots of the equation ${x}^{2}+{x}+{a}=0$ exceed ${a}$, then

  1. $2<{a}<3$
  2. ${a}>3$
  3. $-3<{a}<3$
  4. $ a<-2$
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D Correct answer
Explanation

For roots of x^2 + x + a = 0 to exceed a, the discriminant must be >= 0 (1 - 4a >= 0 => a <= 1/4), the vertex -b/2a = -1/2 must be > a (a < -1/2), and f(a) > 0 (a^2 + a + a > 0 => a^2 + 2a > 0 => a(a+2) > 0). Combining a < -1/2 and a < -2 or a > 0, we get a < -2.

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