Enter coefficients to solve, classify, and graph quadratics. Review roots, axis, vertex, and discriminant quickly. Download polished records and inspect worked example values easily.
General form: f(x) = ax² + bx + c, where a ≠ 0.
Discriminant: D = b² - 4ac.
Roots: x = (-b ± √D) / 2a.
Axis of symmetry: x = -b / 2a.
Vertex: ( -b / 2a , f(-b / 2a) ).
Sum of roots: -b / a.
Product of roots: c / a.
These relationships classify root behavior, locate the turning point, describe graph symmetry, and show how coefficients influence curvature, intercepts, and factorization.
| Polynomial | Coefficients (a, b, c) | Discriminant | Roots Summary | Vertex |
|---|---|---|---|---|
| x² - 3x + 2 | (1, -3, 2) | 1 | 1 and 2 | (1.5, -0.25) |
| 2x² + 4x + 2 | (2, 4, 2) | 0 | Repeated root at -1 | (-1, 0) |
| x² + 2x + 5 | (1, 2, 5) | -16 | -1 ± 2i | (-1, 4) |
The discriminant shows the root type. Positive means two real roots. Zero means one repeated real root. Negative means two complex conjugate roots.
A zero value removes the x² term. The expression then becomes linear, so it no longer represents a quadratic polynomial or parabola.
The vertex is the turning point of the parabola. It gives the minimum value when a is positive, and the maximum value when a is negative.
The axis of symmetry is the vertical line through the vertex. It divides the parabola into mirrored halves and helps locate the graph center quickly.
Yes. When the discriminant is negative, the calculator displays complex conjugate roots in a + bi and a - bi form.
Factor form writes the polynomial using its roots. It is useful for solving equations, checking zeros, and understanding x-intercepts.
The graph helps verify intercepts, opening direction, symmetry, and the vertex. It also makes coefficient comparisons much easier visually.
They include the calculated summary metrics shown in the result table. This makes reporting, classroom sharing, and recordkeeping much easier.
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