Question

If x is real, show that the expression x² + 2x - 11/x - 3 can take all values which do not lie in the open interval (4, 12).

Answer 1

Let

y = x² + 2x - 11/x - 3

Writing this as a quadratic equation in x, we have

x² + x (2 - y) + 3y - 11 = 0 ...(1)

The values of x and y are related by this equation and for each value of y, there is a value of x which is a root of this quadratic equation. In order to that this x (or root) is real, the discriminant ≥ 0.

(2 - y)² - 4(3y - 11) ≥ 0

⇒ y² - 16y + 48 ≥ 0

⇒ (y - 4)(y - 12) ≥ 0

⇒ y ≤ 4 or y ≥ 12

Hence, y (or the given expression) does not take any value between 4 and 12.