Question

Which is the inequality in factored form that represents the region greater than or equal to the quadratic function with zeros –3.5 and 11.5 and includes the point (8.5, –54) on the boundary?

1. The inequality is represented by the polynomial y ≥ 3.6 · (x – 11.5) · (x + 3.5).

### What is the expression of the quadratic inequality?

Herein we must derive the quadratic expression within an inequality of the y ≥ f(x) based on its two roots and a known point on the curve. We can find the coefficients of the quadratic equation by solving this system of linear equations:
(x₁, y₁) = (- 3.5, 0)
12.25 · a – 3.5 · b + c = 0       (1)
(x₂, y₂) = (11.5, 0)
132.25 · a + 11.5 · b + c = 0       (2)
(x₃, y₃) = (8.5, – 54)
72.25 · a + 8.5 · b + c = – 54     (3)
The solution of the system is a = 3.6, b = – 54, c = 144.9. Thus, the inequality is represented by the polynomial y ≥ 3.6 · x² – 54 · x + 144.9, whose factored form is determined by the quadratic formula:
y ≥ 3.6 · (x² – 15 · x + 40.25)
y ≥ 3.6 · (x – 11.5) · (x + 3.5)