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Löser der quadratischen Formel

EntwicklerMathe

[iotools_quadratic_formula_solver]

Führung

Löser der quadratischen Formel

Enter the coefficients le, bund c of any quadratic equation in the form ax² + bx + c = 0 and instantly get the discriminant, the two roots (real or complex), the vertex, the axis of symmetry, the vertex form, a step-by-step derivation, and a visual sketch of the parabola. Everything is computed locally in your browser using exact closed-form math — no rounded approximations, no AI guesses, no hidden API calls.

Nutzung

Type the leading coefficient le (must not be zero for a quadratic).
Type the linear coefficient b (use a minus sign for negative values, e.g. -3).
Type the constant c.
The summary, step-by-step solution, and parabola sketch update as you type.
Click any of the quick examples to load a two-real-roots, repeated-root, complex-roots, or golden-ratio case.
Press the copy button to grab the full plain-text summary for notes or homework.

Funktionen

Discriminant breakdown – Computes Δ = b² − 4ac and labels the case (two real, repeated, or complex).
All root types – Handles two distinct real roots, one repeated real root, and complex conjugate roots in a ± bi format.
Vertex and axis of symmetry – Shows the vertex (h, k), the axis x = h, and whether the parabola opens upward or downward.
Vertex form – Rewrites the quadratic as a(x − h)² + k for graphing or transformation work.
Full step-by-step – Shows every substitution from the discriminant to the final roots so you can check your homework or learn the method.
SVG parabola sketch – Renders a clean, scalable parabola with vertex, real roots, and axis of symmetry highlighted.
Edge cases handled – Detects degenerate inputs (a = 0 falls back to a linear equation; a = b = 0 reports the trivial cases).
Private and offline-capable – All math runs in your browser. No coefficients are ever sent to a server.

 Häufig gestellte Fragen

What is the quadratic formula?

The quadratic formula gives the roots of any equation of the form ax² + bx + c = 0 with a ≠ 0. It is x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, b² − 4ac, is called the discriminant and decides whether the roots are real or complex.
What does the discriminant tell you?

The discriminant Δ = b² − 4ac classifies the roots without computing them. If Δ is greater than zero the parabola crosses the x-axis at two distinct real points. If Δ equals zero the parabola just touches the x-axis at a single repeated root. If Δ is less than zero the parabola does not cross the x-axis and the two roots are complex conjugates of the form −b/(2a) ± i√|Δ|/(2a).
What are complex roots and when do they appear?

Complex roots appear whenever the discriminant is negative, because you cannot take the real square root of a negative number. The roots are always written as a conjugate pair p + qi and p − qi, where i is the imaginary unit defined by i² = −1. Geometrically this means the parabola never intersects the x-axis at any real point.
What is the vertex form of a quadratic and why is it useful?

The vertex form a(x − h)² + k is obtained by completing the square on the standard form. It exposes the vertex (h, k) directly, so you can read off the parabola's lowest point (when a is positive) or highest point (when a is negative) without solving the equation. It also makes graphing and translation problems much easier.
What is the axis of symmetry of a parabola?

Every parabola is symmetric about a vertical line that passes through its vertex. For y = ax² + bx + c that line is x = −b/(2a), the same value as the x-coordinate of the vertex. Knowing the axis of symmetry lets you mirror any point on the parabola to find a matching point on the opposite side.

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