Quadratics & Nonlinear Functions
Quadratics can be solved by factoring, the quadratic formula, or completing the square; understanding vertex form and the discriminant helps you quickly answer questions about a parabola's graph.
Solving and interpreting quadratics
Factoring is fastest when possible: find two numbers that multiply to give ac and add to give b (for ax²+bx+c). The quadratic formula x = [-b ± √(b²-4ac)] / 2a always works as a backup.
The discriminant b²-4ac tells you the number of real solutions: positive → two real solutions, zero → exactly one (repeated) real solution, negative → no real solutions.
Vertex form y = a(x-h)² + k directly shows the vertex (h,k) — useful for max/min word problems without any calculus.
Worked Example
Solve x² - 5x + 6 = 0 by factoring.
- Find two numbers that multiply to 6 and add to -5: -2 and -3.
- Factor: (x-2)(x-3) = 0.
- Set each factor to 0: x=2 or x=3.
Answer: x = 2 or x = 3
Key terms (2)
- Discriminant
- b²-4ac; tells the number of real solutions to a quadratic.
- Vertex form
- y = a(x-h)² + k, where (h,k) is the vertex of the parabola.
Practice Quiz
How many real solutions does 2x² + 3x + 5 = 0 have?