Multiple choice

Let $S$ be the set of values of $a$ for which $2$ lies between the roots of the quadratic equation $x^{2} + (a + 2)x - (a + 3) = 0$, then $S$ is given by

  1. $(-\infty , -5]$
  2. $(5 , \infty)$
  3. $(-\infty , -5)$
  4. $[5 , \infty)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For 2 to lie between the roots of x^2 + (a + 2)x - (a + 3) = 0, the value of the function f(x) = x^2 + (a + 2)x - (a + 3) at x = 2 must be negative. Substituting x = 2 gives 4 + 2(a + 2) - (a + 3) = 4 + 2a + 4 - a - 3 = a + 5. Setting a + 5 < 0 yields a < -5.

AI explanation

To ensure the value 2 lies between the roots of the quadratic equation x^2 + (a + 2)x - (a + 3) = 0, we use the standard condition for a number k to lie strictly between the roots: f(k) must be less than 0. The leading coefficient is positive (1), so we evaluate f(2). Substituting x = 2 gives (2)^2 + (a + 2)(2) - (a + 3), which simplifies to 4 + 2a + 4 - a - 3, equaling a + 5. For 2 to lie between the roots, we must have a + 5 < 0. Solving this inequality gives a < -5, defining the set S as (-infinity, -5).

Keep practicing — related questions