Quadratic Equation Solver & Grapher

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Category Math & Algebra

Equation Coefficients

ax² + bx + c = 0

a (Quadratic)

b (Linear)

c (Constant)

Discriminant (Δ): -

Root 1 (x₁): -

Root 2 (x₂): -

Vertex: -

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About

Quadratic equations model parabolic trajectories, fluid dynamics, and profit maximization. Solving them manually involves the quadratic formula, but visualizing the result provides deeper intuition. This tool combines a logic engine for finding roots - including imaginary numbers (i) when the parabola never touches the X-axis - with a vector-based graphing engine.

The solver computes the discriminant Δ to determine the nature of the roots. If Δ < 0, the solutions exist in the complex plane. The accompanying graph dynamically scales the viewport to focus on the vertex and intercepts, ensuring the curve is always visible.

Formulas

The standard form is ax² + bx + c = 0. The roots are found via:

x = −b ± √b² − 4ac2a

The vertex coordinates (h, k) are calculated as:

h = −b2a , k = c − b²4a

Reference Data

Discriminant (Δ)	Root Type	Graph Behavior
Δ > 0	Two distinct real roots	Intersects X-axis twice
Δ = 0	One real repeated root	Touches X-axis at vertex
Δ < 0	Two complex conjugate roots	Does not touch X-axis

Frequently Asked Questions

If the coefficient a is 0, the equation is no longer quadratic; it becomes linear (bx + c = 0). The graph becomes a straight line, and there is only one root at x = -c/b.

The discriminant is the value inside the square root: b² - 4ac. It tells you how many solutions exist. Positive means 2 real solutions, zero means 1 solution, and negative means 2 imaginary solutions.

Complex roots imply the parabola does not intersect the real X-axis. The graph will either float above the axis (if a > 0) or sit below it (if a < 0). The roots are not visually plottable on a standard 2D Cartesian real plane.