Multiple choice

If $\displaystyle 1 \in (\alpha, \beta )$ where $\alpha,\ \beta$ are the roots of the equation $\displaystyle x^{2}-a(x+1)+3=0$, then

  1. $\displaystyle a>2$
  2. $\displaystyle a<-6 \:or \:a>2$
  3. $\displaystyle - 6 < a < 2 $
  4. $\displaystyle a<2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let f(x) = x^2 - a(x+1) + 3. If 1 is between roots, f(1) < 0. f(1) = 1 - a(2) + 3 = 4 - 2a. 4 - 2a < 0 => 2a > 4 => a > 2.

AI explanation

If one lies between the roots alpha and beta of the quadratic equation, the value of the function at x equals one must have the opposite sign of the x squared coefficient. Substituting x equals one into x squared minus a times the quantity x plus one plus three gives one minus two a plus three, which equals four minus a. Since the x squared coefficient is positive, four minus a must be less than zero, meaning a is greater than two.

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