Key #1
DISCRIMINANT
Tells you HOW MANY real roots exist. Always check this first!
Δ = b² − 4ac
Δ > 0 → 2 roots · Δ = 0 → 1 root · Δ < 0 → no real roots
Key #2
VERTEX FORM
Vertex is directly visible. Perfect for min/max questions.
y = a(x − h)² + k
Vertex = (h, k) · Axis of symmetry: x = h
Key #3
VIETA'S FORMULAS
Sum & product of roots without solving. SAT LOVES this!
r₁+r₂ = −b/a · r₁·r₂ = c/a
Key #4
COMPLETE THE SQUARE
Convert standard → vertex form. Find vertex fast.
x² + bx = (x + b/2)² − (b/2)²
Key #5
NO SOLUTION RULE
System of line + parabola has NO real solution when:
Discriminant of combined eq < 0
Key #6
AXIS OF SYMMETRY
Midpoint of the two x-intercepts. From standard form:
x = −b / (2a)