How this calculator works
Find the real roots to split the number line into sign regions. An upward parabola is positive outside two roots and negative between them; a downward parabola reverses these signs. Inclusive comparisons include zero-valued roots. A repeated root touches the axis without changing sign. Without real roots, the sign is the sign of a everywhere.
Formula
D = b^2 - 4ac. For two roots r1 < r2 and a > 0, P(x) < 0 on (r1,r2) and P(x) > 0 outside. Reverse signs for a < 0.
Worked example
x^2 - 5x + 6 < 0 factors as (x - 2)(x - 3) < 0, giving (2, 3). For -x^2 + 4 <= 0, the solution is (-inf, -2] U [2, inf).
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Frequently asked questions
Do inclusive comparisons always give an interval of positive length?
No. For (x - 2)^2 <= 0, only x = 2 works, written [2, 2].
What if there are no real roots?
The quadratic keeps one sign on the entire real line. The solution is all real numbers or the empty set.
Can this draw a parabola?
The readout gives direction and the solution intervals. The current engine does not draw a curve or shaded graph.