Quadratic Equation Solver
Quickly solve any quadratic of the form ax² + bx + c = 0 and see whether the roots are real or complex from the discriminant. It’s useful for algebra homework checks, modeling parabolic motion in science, or optimizing curved relationships in economics and engineering when factoring is slow or uncertain.
The Quadratic Equation Solver takes your coefficients and applies the quadratic formula to deliver clear, reliable solutions while highlighting what the discriminant and vertex mean for the parabola’s shape. The goal is fast, error-free answers that build intuition about symmetry, turning points, and intercepts—so you can move from setup to insight with confidence.
Solve ax² + bx + c = 0 with exact and decimal roots, discriminant, vertex, axis of symmetry, intercepts, and a mini-graph.
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Quadratic Equation Results
How we solve ax² + bx + c = 0
Quadratic formula, vertex, and intercepts
Quadratic formula: x = (-b ± √(b² − 4ac)) / (2a). The discriminant D = b² − 4ac decides the root type.
Root types: if D > 0 → two real roots; D = 0 → one real double root; D < 0 → complex roots.
Vertex: h = −b/(2a), k = f(h).
Axis of symmetry: x = h. Intercepts: y-intercept at (0, c); x-intercepts are the roots (if real).
Exact vs decimal: we keep √D in exact mode unless D is a perfect square; otherwise we show decimals.
Assumptions: a ≠ 0 for a true quadratic. We detect linear a=0 and show the linear solution.
Results interpretation: when this is useful
- Algebra practice: compare hand-worked steps against exact and decimal forms.
- Graph reading: verify vertex and intercepts before sketching.
- Parameter intuition: adjust a, b, c to see how opening, shift, and width change the curve.
Use cases & examples
Example 1: a = 1, b = −3, c = 2 → D = 1 → roots 1 and 2; vertex (1.5, −0.25).
Example 2: a = 2, b = 4, c = 2 → D = 0 → double root x = −1; vertex (−1, 0).
Example 3: a = 1, b = 2, c = 5 → D = −16 → complex roots −1 ± 2i; vertex (−1, 4).
Quadratic Solver — FAQ
What does the discriminant tell us?
D = b² − 4ac determines root type: D > 0 → two distinct real roots, D = 0 → one real double root, D < 0 → complex roots.
How do you find the vertex of a parabola?
Use h = −b/(2a) and k = f(h) = a·h² + b·h + c. The axis of symmetry is x = h.
Why are there exact and decimal roots?
Exact roots preserve radicals like √D for symbolic work. Decimal roots are convenient approximations for quick estimates.
What happens if a = 0?
The expression becomes linear (bx + c = 0). We detect this case and show the linear solution x = −c/b.
How do I know if the parabola crosses the x-axis?
Check the discriminant: D > 0 crosses twice, D = 0 touches once, D < 0 never crosses.
What does coefficient a change about the graph?
It sets opening direction and width. a > 0 opens up, a < 0 opens down. Larger |a| makes the parabola narrower.
Understanding quadratics: structure, shape, and strategy
We treat ax² + bx + c as a shape with a predictable backbone: the vertex fixes location, the coefficient a sets opening and scale, and the discriminant forecasts how the curve meets the axis. Our goal is clarity—transparent algebra, readable numbers, and a quick plot to tie it together.
From coefficients to geometry
The term a stretches the parabola vertically; larger |a| narrows the curve. The sign of a sets the opening direction. The linear term b shifts the vertex horizontally via −b/(2a), while c pins the y-intercept at (0, c).
Why the discriminant matters
D = b² − 4ac gathers the effect of all three coefficients into a single forecast about the roots. Positive D means two crossings, zero means a gentle tangent, and negative D means the curve floats above or below the axis without crossing.
A practical workflow
We encourage setting a, b, c; scanning D; locating the vertex; and then using the intercepts for a quick sketch. Switching between exact and decimal views reinforces symbolic fluency and numerical intuition.