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Discriminant Quick Reference
- Δ = b² - 4ac
- Δ > 0: Two distinct real roots — parabola crosses x-axis at two points
- Δ = 0: One repeated real root — parabola touches x-axis at one point (vertex)
- Δ < 0: Two complex conjugate roots — parabola does not intersect x-axis
- Quadratic formula: x = (-b ± √Δ) / 2a
- Vertex: x = -b/2a, y = c - b²/4a
- Axis of symmetry: x = -b/2a
Examples
Example 1: Two Distinct Real Roots
Equation: x² - 5x + 6 = 0
a = 1, b = -5, c = 6
Step 1 — Compute discriminant:
Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
Step 2 — Discriminant > 0 → two distinct real roots
Step 3 — Apply quadratic formula:
x = (-b ± √Δ) / 2a
x = (5 ± √1) / 2
x = (5 ± 1) / 2
x₁ = (5 + 1) / 2 = 3
x₂ = (5 - 1) / 2 = 2
Roots: x = 3 and x = 2
Factored form: (x - 3)(x - 2) = 0
Vertex: (2.5, -0.25)
Example 2: One Repeated Root (Perfect Square)
Equation: x² - 6x + 9 = 0
a = 1, b = -6, c = 9
Step 1 — Compute discriminant:
Δ = (-6)² - 4(1)(9) = 36 - 36 = 0
Step 2 — Discriminant = 0 → one repeated root
Step 3 — Apply formula:
x = -b / 2a = 6 / 2 = 3
Root: x = 3 (repeated)
Factored form: (x - 3)² = 0
Vertex form: (x - 3)² = 0
Vertex: (3, 0) — parabola touches x-axis at exactly one point
Example 3: Complex Roots (No Real Solutions)
Equation: x² + 2x + 5 = 0
a = 1, b = 2, c = 5
Step 1 — Compute discriminant:
Δ = 2² - 4(1)(5) = 4 - 20 = -16
Step 2 — Discriminant < 0 → two complex conjugate roots
Step 3 — Apply formula:
x = (-2 ± √(-16)) / 2
x = (-2 ± 4i) / 2
x₁ = -1 + 2i
x₂ = -1 - 2i
Roots: x = -1 + 2i and x = -1 - 2i
Geometric meaning: The parabola y = x² + 2x + 5
does not intersect the x-axis (minimum value is 4,
which is above the x-axis).