Study notes
Quadratic Functions & Inequalities
Completing the square to find the minimum/maximum and range of a quadratic function, using the discriminant to determine the nature of roots (including tangency/intersection problems), and solving quadratic inequalities by sketching or sign analysis.
What this topic covers
- Completing the square turns \(ax^2+bx+c\) into \(a(x-p)^2+q\)
- The completed-square form gives the minimum or maximum value directly
- The discriminant \(b^2-4ac\) tells you the nature of the roots
- The discriminant decides whether a line touches, crosses, or misses a curve
- Solve a quadratic inequality by sketching the parabola, not by solving the equation alone
- Sign analysis is a quick alternative to sketching
- Match strict and non-strict inequality signs carefully
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