Solve ax² + bx + c = 0 with real and complex roots, discriminant analysis, vertex coordinates, and step-by-step completing the square.
Click "Calculate" to solve equation.
A ball thrown into the air follows a parabolic arc modeled by h(t) = -4.9t² + 20t + 1.5. Finding when the ball hits the ground requires solving for t when h(t) = 0. A civil engineer designing suspension bridge cables uses quadratic equations to determine cable sag and structural tension points. An economics team modeling profit functions P(x) = -2x² + 120x - 1000 solves for break-even production levels where profit equals zero.
A quadratic equation is a second-order polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. The term a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. The graph of a quadratic function is a U-shaped curve called a parabola. The points where the parabola crosses the x-axis are the roots or solutions of the equation.
The standard method for solving any quadratic equation is the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). The expression inside the square root, b² - 4ac, is known as the discriminant (symbolized by Δ). The discriminant determines whether the roots are real numbers, repeated real numbers, or complex imaginary conjugate pairs. The following guide covers the algebraic methods, step-by-step algorithms, and practical applications of quadratic equations.
When you click "Solve Equation," the JavaScript engine reads the coefficients a, b, and c. It computes the discriminant Δ and selects the appropriate algebraic pathway.
1. Discriminant Calculation: Δ = b² - 4ac
2. Root Classification:
x₁ = (-b + √Δ) / (2a)x₂ = (-b - √Δ) / (2a)
x = -b / (2a)
u = -b / (2a)v = √(|Δ|) / (2a)x₁ = u + vi, x₂ = u - vi
3. Vertex Coordinates: The vertex (h, k) represents the peak or trough of the parabola:
h = -b / (2a)
k = c - (b² / (4a))
4. Completing the Square Algorithm:
Divide by a: x² + (b/a)x = -c/a
Add (b / 2a)² to both sides: (x + b/2a)² = (b² - 4ac) / (4a²)
Take square root: x + b/2a = ±√(b² - 4ac) / (2a)
Isolate x: x = (-b ± √(b² - 4ac)) / (2a)
Physics and projectile motion. Trajectories of thrown objects, rockets, and water jets follow parabolic equations y = y₀ + v₀t - ½gt². Solving quadratic equations determines total flight duration and splashdown times.
Architecture and structural engineering. Arch bridges, parabolic satellite dishes, and acoustic reflectors use quadratic curves to focus wave energy and distribute weight efficiently.
Business and revenue optimization. Revenue curves R(p) = p × (m - np) = mp - np² are quadratic. Finding the vertex identifies the price point p that maximizes total business revenue.
Computer graphics and 3D ray tracing. Ray-sphere intersection algorithms solve quadratic equations to compute where light rays intersect 3D spherical geometry in computer games and CGI animation.
Land surveying and area calculations. Determining dimensions of rectangular land parcels when given perimeter and total square footage leads directly to quadratic equations.
Ensure coefficient a is non-zero. If a = 0, the equation is linear (bx + c = 0), which will trigger a validation alert.
Pay attention to negative signs. If your equation is x² - 5x + 6 = 0, enter a = 1, b = -5, and c = 6.
For solving systems of linear equations alongside quadratic models, use our System of Equations Solver. For evaluating exponential growth or logarithmic curves, pair this tool with our Logarithm Calculator.
The solver runs locally in JavaScript using double-precision floating-point arithmetic. Roots and vertex coordinates evaluate in under 1 millisecond.
| Feature | This Tool | Hand Quadratic Formula | Graphing Calculator |
|---|---|---|---|
| Real Roots | Instant (<1ms) | 2-5 minutes | Instant |
| Complex Roots | Exact (u ± vi) | Manual i substitution | Requires complex mode |
| Discriminant Δ | Automatic display | Manual calculation | Not explicitly listed |
| Vertex Form (h, k) | Automatic conversion | Manual completion | Trace function |
| Privacy | Client-side browser | Paper | Local device |
| Cost | Free | Free | Hardware purchase |
When Δ < 0, the parabola does not cross the x-axis. The roots are complex numbers containing the imaginary unit i (where i = √(-1)).
Calculate h = -b / (2a) and k = c - (b² / (4a)). The vertex form is a(x - h)² + k.
The axis of symmetry is the vertical line passing through the vertex, given by the equation x = -b / (2a).
When Δ = 0, the vertex of the parabola touches the x-axis at exactly one point, producing a double or repeated real root.
Yes. You can enter fractional or decimal coefficients like a = 0.5, b = -2.4, c = 1.8.
System of Equations Solver — Solves 2×2 and 3×3 linear systems using Gaussian elimination.
Scientific Calculator — Evaluates complex expressions, trigonometric functions, and logarithms.
Matrix Calculator — Computes determinants, inverses, and matrix products.