Mastering Quadratic Equations in 2026
Everything you need to know about the quadratic formula, vertex form, and factoring.
The Quadratic Formula
The universal method for solving any quadratic equation.
- • a, b, c: Coefficients from ax² + bx + c
- • ±: Indicates two potential solutions
The Vertex Form
Best for graphing and finding max/min values.
- • (h, k): Coordinates of the vertex
- • h: Axis of symmetry (-b/2a)
Example Quadratic Equations (2026)
The table below shows example quadratic equations with their coefficients, discriminant values, and solutions. These examples demonstrate different scenarios: two real roots, one repeated root, and complex roots.
| Equation | a | b | c | Discriminant (Δ) | Roots | Type |
|---|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 | 1 | x = 2, x = 3 | Two real |
| x² - 4x + 4 = 0 | 1 | -4 | 4 | 0 | x = 2 (repeated) | One real |
| x² + 2x + 5 = 0 | 1 | 2 | 5 | -16 | x = -1 ± 2i | Complex |
| 2x² - 7x + 3 = 0 | 2 | -7 | 3 | 25 | x = 0.5, x = 3 | Two real |
| 3x² + 6x + 3 = 0 | 3 | 6 | 3 | 0 | x = -1 (repeated) | One real |
| x² + x + 1 = 0 | 1 | 1 | 1 | -3 | x = -0.5 ± 0.866i | Complex |
The discriminant (Δ = b² - 4ac) determines the nature of roots: Δ > 0 (two real roots), Δ = 0 (one repeated root), Δ < 0 (two complex roots).
Understanding the Discriminant (Δ)
The term inside the square root, b² - 4ac, is called the discriminant. It acts as a predictor for the type of solutions you will get:
How to Convert Standard Form to Vertex Form
Converting from Standard Form (ax² + bx + c) to Vertex Form (a(x-h)² + k) is a crucial skill in algebra.
- Find 'h' (x-coordinate of vertex):h = -b / (2a)
- Find 'k' (y-coordinate of vertex):k = c - (b² / 4a)Alternatively, plug 'h' back into the original equation: k = f(h).
- Write the equation:Substitute a, h, and k into y = a(x - h)² + k.
Usage in Real Life
- 🚀Physics & Ballistics
Calculating the trajectory of a projectile, finding max height, or time of flight.
- 💰Economics
Finding maximum profit or minimum cost when the relationship is non-linear.
- 🌉Engineering
Designing parabolic structures like bridges, satellite dishes, and headlight reflectors.
Frequently Asked Questions
- Can I solve quadratic equations with complex numbers?
- Yes! Our calculator fully supports complex numbers. If the discriminant is negative, the solution will include the imaginary unit i.
- What is the difference between Roots and Zeros?
- They are essentially the same thing. "Roots" usually refers to the solutions of an equation (e.g., ax² + bx + c = 0), while "Zeros" refers to the x-values where a function equals zero (e.g., f(x) = 0).
- Why is the Vertex Form useful?
- Vertex Form immediately gives you the turning point (vertex) of the parabola. This is critical for optimization problems where you need to find the maximum or minimum value.