The formula
ax² + bx + c ⋛ 0
Why it works
An inequality looks like an equation but behaves differently at one crucial point: multiplying or dividing both sides by a negative number reverses the direction. It is the single most repeated mistake in school algebra. For a quadratic there is no mechanical rule but a geometric argument: the roots split the real line into stretches, and the sign of the parabola is constant inside each one. If it opens upwards it is negative between the roots and positive outside; if it opens downwards, the other way round. All you need is which way it opens and where it crosses.
How to solve it by hand
- Move everything to one side so the inequality is compared with zero
- For a linear one, isolate x and flip the sign if you divide by a negative
- For a quadratic, find the roots: they split the line into stretches
- See which way the parabola opens and pick the stretches that satisfy it
What is worth knowing
The discriminant settles the case before anything is calculated. If it is negative the parabola never crosses the axis and its sign is that of a everywhere: the inequality holds always or never, and there is no interval to give. If it is zero the parabola touches the axis without crossing, and there the difference between strict and non-strict changes the answer completely: (x−1)² > 0 holds for every number except one, while (x−1)² ≥ 0 holds for all of them. That is the detail most often missed in exams.