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Sign of Quadratic Function

Let y = f(x) = ax2 + bx + c, where a, b, c, R and a ≠ 0.
∴ 62413.png
where D is discriminant.
 
The graph of y = f(x) is parabola whose axis is parallel to the y-axis and vertex A62407.png.
 
For some values of x, f(x) may be positive, negative, or zero and for a > 0, the parabola opens upwards and for a < 0, the parabola opens downwards, this gives the following cases:
 
a > 0 and D < 0, then f(x) > 0 x R
62393.png
a > 0 and D > 0
62386.png
a > 0 and D = 0, then f(x) ≥ 0 x R
62380.png
a < 0 and D < 0 then f(x) < 0 x R
62374.png
a < 0 and D > 0
62368.png
a < 0 and D = 0 then f(x) ≤ 0 x R
62362.png
 
Note: If f(x) ≥ 0, ∀ x ∈ R ⇒ a > 0 and D ≤ 0 and if f(x) ≤ 0 ∀ x ∈ R ⇒ a < 0 and D ≤ 0.




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