Quadratic Formula Calculator
Welcome to Calculator Kilo's Quadratic Formula Calculator
Your fast-track to solving any quadratic equation of the form ax² + bx + c = 0. Whether you're double-checking homework, modeling projectile motion, or coding a physics engine, our calculator returns every valid root instantly while showing the discriminant, vertex, and axis of symmetry.
Why use our calculator?
- Instantly calculates all valid roots
- Displays discriminant, vertex and axis of symmetry
- Handles complex numbers and fractions
- Shows step-by-step solutions
1. Quadratic Equations at a Glance
| Symbol | Meaning | Why it Matters |
|---|---|---|
| a | Quadratic (squared-term) coefficient | Shapes the parabola (opens up if a > 0, down if a < 0) |
| b | Linear coefficient | Tilts/shifts the curve left or right |
| c | Constant | Vertical intercept of the graph |
| Δ = b² – 4ac | Discriminant | Determines number and nature of roots |
Discriminant Decision Table
| Δ Value | Nature of Roots | Graph Behaviour |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One real (repeated) root | Vertex touches x-axis |
| Δ < 0 | Two complex roots | Never intersects x-axis |
2. The Quadratic Formula
This single expression works for every quadratic equation. Just enter your a, b, and c values to instantly receive:
- Both roots (x₁ and x₂)
- Discriminant Δ
- Vertex (h, k) where h = -b/(2a)
- Axis of symmetry x = h
3. From Formula to Graph
Consider the equation: y = x² - 5x + 6
- Roots at x = 2 and x = 3 (where parabola intersects x-axis)
- Vertex halfway between roots at (2.5, -0.25)
- Axis of symmetry at x = 2.5 (vertical line)
Seeing algebra turn into geometry makes the math stick!
4. Quick Walk-Through: Completing the Square
- Divide every term by a
- Isolate x² + (b/a)x and add (b/2a)² to both sides
- Factor the perfect square: (x + b/2a)²
- Solve for x by taking square roots and rearranging
Follow these steps to re-derive the quadratic formula yourself!
5. Calculator How-To
- Enter coefficients (a, b, c)
- Click Calculate
- Review results in the output card
- Toggle "Show work" for step-by-step solutions
- Click Reset for new problems
6. Real-World Applications
- Physics: Projectile motion calculations
- Engineering: Arch and bridge design
- Business: Profit curve analysis
- Computer Graphics: Collision detection in games
7. Frequently Asked Questions
Q: Can the calculator handle complex roots?
A: Yes - solutions appear in a ± bi format.
Q: What about decimals or fractions?
A: Enter values like 0.75 or 3/4 - automatic conversion included.
Q: Why does my equation return an error?
A: Check that a ≠ 0 and all fields contain valid numbers.
Solve smarter, learn faster, and let the math curve in your favour with Calculator Kilo's Quadratic Formula Calculator.