In-depth
Quadratic Equations: Theory and Solution
A quadratic equation (or second-degree equation) is an algebraic equation in which the highest degree of the unknown is 2. It is one of the pillars of algebra and appears in countless applications: from physics to trajectory calculations, from economics to engineering.
Standard Form
Every quadratic equation can be written in standard form:
ax² + bx + c = 0
where a, b, and c are real coefficients and a ≠ 0 (if a were zero, the equation would be linear).
The Discriminant
The key to solving a quadratic equation is the discriminant (denoted by the Greek letter delta):
Δ = b² – 4ac
The value of the discriminant determines the number and nature of the solutions:
| Value of Δ | Real Solutions | Description |
|---|---|---|
| Δ > 0 | 2 distinct | The parabola intersects the x-axis at two points |
| Δ = 0 | 1 (repeated) | The parabola is tangent to the x-axis |
| Δ < 0 | 0 (2 complex) | The parabola does not intersect the x-axis |
The Quadratic Formula
The solutions (roots) of the equation are obtained using the quadratic formula:
x = (–b ± √Δ) / (2a)
which, expanding the discriminant, becomes:
x = (–b ± √(b² – 4ac)) / (2a)
Examples for Each Case
Case 1: Δ > 0 — Two Distinct Real Solutions
Solve x² – 5x + 6 = 0 (a=1, b=–5, c=6)
- Δ = (–5)² – 4(1)(6) = 25 – 24 = 1
- x₁ = (5 + 1) / 2 = 3
- x₂ = (5 – 1) / 2 = 2
Case 2: Δ = 0 — Repeated Solution
Solve x² – 6x + 9 = 0 (a=1, b=–6, c=9)
- Δ = (–6)² – 4(1)(9) = 36 – 36 = 0
- x = 6 / 2 = 3 (repeated solution)
Case 3: Δ < 0 — No Real Solutions
Solve x² + 2x + 5 = 0 (a=1, b=2, c=5)
- Δ = (2)² – 4(1)(5) = 4 – 20 = –16
- No real solutions exist
- Complex solutions: x = (–2 ± 4i) / 2 → x₁ = –1 + 2i, x₂ = –1 – 2i
Vertex of the Parabola
The equation ax² + bx + c = 0 is associated with the parabola y = ax² + bx + c. The vertex of the parabola has coordinates:
V = (–b/(2a), –Δ/(4a))
The vertex represents the minimum point (if a > 0) or the maximum point (if a < 0) of the parabola. For the equation x² – 5x + 6 = 0:
- xᵥ = 5 / 2 = 2.5
- yᵥ = –1 / 4 = –0.25
- Vertex: V(2.5, –0.25)
Relationships Between Roots and Coefficients (Vieta's Formulas)
If x₁ and x₂ are the roots of the equation, the following relationships hold:
- x₁ + x₂ = –b/a (sum of roots)
- x₁ × x₂ = c/a (product of roots)
These formulas allow you to quickly verify the solutions found and are useful for constructing equations from desired roots.
Special Cases
| Type | Condition | Quick Method |
|---|---|---|
| Pure | b = 0 → ax² + c = 0 | x = ±√(–c/a) |
| Incomplete | c = 0 → ax² + bx = 0 | x(ax + b) = 0 → x=0, x=–b/a |
| Trivial | b = 0 and c = 0 | x = 0 (repeated) |
How to Use the Calculator
Enter the coefficients a, b, and c of the equation in standard form. The solver will calculate the discriminant, the solutions (real or complex), the coordinates of the parabola's vertex, and verify the results using Vieta's formulas. Decimal and negative values are accepted for the coefficients.
