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Quadratic equations

Quadratic Formula Calculator

Solve ax² + bx + c = 0. Roots in simplified radical and decimal form, discriminant, vertex, parabola. Complex roots when D < 0. Free step-by-step quadratic solver.

Quadratic Intelligence Lab

Enter coefficients for ax² + bx + c = 0. Roots are shown in simplified radical form and decimal form. Complex roots use i when D < 0.

Must be non-zero

Solutions

Roots x₁ and x₂

x₁ = 1.5 + 0.5

x₂ = 1.5 − 0.5

Decimal: 2.000000 and 1.000000

Discriminant D1b² − 4ac
Vertex (h, k)(1.500000, -0.250000)x = −b/(2a)
Y-intercept (0, c)(0, 2)

Parabola f(x) = ax² + bx + c

x₁x₂V: 1.5, -0.25Y: 0, 2

Logic trace

  • Step 1: Identify a, b, ca = 1, b = -3, c = 2
  • Step 2: Discriminant D = b² − 4acD = -3² − 4(1)(2) = 9 − 8 = 1
  • Step 3: Substitute into x = (−b ± √D) / (2a)x = (−(-3) ± √1) / (2(1)) = 1.5 + 0.5 and 1.5 − 0.5

How to Use This Calculator

Enter a, b, and c for ax² + bx + c = 0. Roots, parabola, and Logic Trace update automatically. Use the graph to see vertex, axis of symmetry, and roots (or the complex-roots note).

Workflow Tips

Enter Coefficients

Use any real numbers; a must be non-zero. You get x₁ and x₂, D, vertex, and y-intercept.

Real vs Complex Roots

D ≥ 0: two real roots (or one repeated). D < 0: roots shown with i; the parabola does not cross the x-axis.

Graph Labels

Green points = roots on the x-axis. Amber = vertex. primary = y-intercept (when x = 0 is in view). Dashed line = axis of symmetry.

Copy Results

Copies the equation, formula, D, roots (radical and decimal), vertex, and y-intercept.

Quadratic Formula Calculator: Roots, Discriminant & Parabola Graph

Free quadratic formula calculator: solve ax² + bx + c = 0. Find roots in simplified radical form and decimal form. Discriminant D = b² − 4ac, vertex (h,k), parabola with axis of symmetry. Complex roots when D < 0. How to solve quadratic equations; quadratic equation solver for algebra and math. Trusted by students and educators. No sign-up.

What This Calculator Does & Who It's For

Calculator Purpose & Ideal Users

This quadratic formula calculator solves ax² + bx + c = 0 and is widely used in high school algebra and college math. It shows roots in simplified radical form (e.g. 2 ± 3√5) and decimal form, the discriminant, vertex, and a parabola graph with axis of symmetry and labeled points.
  • What You'll Get:
    Roots x₁ and x₂ in radical and decimal form; D = b² − 4ac; vertex (h, k) and y-intercept (0, c); a parabola visualizer with vertex, axis of symmetry, roots (or “no real x-intercepts” when D < 0), and y-intercept when in view; a logic trace (identify a, b, c; compute D; substitute into x = (−b ± √D)/(2a)). When D < 0, roots use the imaginary unit i (e.g. 1 ± 2i).
  • Ideal Users:
    Students & algebra: Solve quadratic equations, get radical form and decimals, check homework. Teachers: Demonstrate the formula, discriminant, and parabola geometry. Anyone: Quick quadratic equation solver with clear steps and visuals. How to solve a quadratic equation: enter a, b, c and read the roots and graph.
  • Scope & Limits:
    Real coefficients only. Coefficient a must be non-zero. All calculations run locally; no sign-up.

How to Use This Calculator

Enter the three coefficients a, b, and c for the equation ax² + bx + c = 0. Coefficient a must be non-zero; b and c accept any real number, including zero and negatives. Results update instantly as you type: the discriminant, both roots, vertex coordinates, and the parabola graph all refresh in real time.
  • Discriminant (D):
    Shown at the top of the results panel. D > 0 means two distinct real roots; D = 0 means one repeated root; D < 0 means two complex conjugate roots displayed with the imaginary unit i.
  • Roots:
    Each root appears in simplified radical form (e.g. 2 ± 3√5) and as a decimal approximation. Complex roots show the real and imaginary parts separately.
  • Parabola Graph:
    The graph plots f(x) = ax² + bx + c with labeled points: roots in green on the x-axis, vertex in amber, y-intercept when visible, and a dashed axis of symmetry at x = −b/(2a).
  • Copy Results:
    Click the Copy button to export the equation, formula substitution, discriminant, roots, vertex, and y-intercept as formatted text for homework or documentation.

The Quadratic Formula and Discriminant

For ax² + bx + c = 0 with a ≠ 0, the roots are
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The discriminant is
D=b2−4acD = b^2 - 4ac
If D > 0, there are two distinct real roots. If D = 0, one repeated (double) root. If D < 0, the roots are complex conjugates: x = (−b ± i√|D|) / (2a). This quadratic formula calculator shows all three cases with simplified radical form (√D simplified, e.g. 2√5) and decimal approximations.

Vertex and Axis of Symmetry of the Parabola

The parabola f(x) = ax² + bx + c has vertex (h, k) where h = −b/(2a) and k = f(h). The axis of symmetry is the vertical line x = h. The calculator displays the vertex and draws the axis on the graph. When D < 0, the parabola does not cross the x-axis (no real roots); the graph shows the curve and a note that roots are complex.

How to Solve a Quadratic Equation Step by Step

To solve a quadratic equation using the formula: (1) Write the equation as ax² + bx + c = 0 and identify a, b, and c. (2) Compute D = b² − 4ac. (3) Substitute into x = (−b ± √D) / (2a); if D < 0, use √D = i√|D|. (4) Simplify radicals and decimals. This quadratic equation solver does steps 2–4 for you and shows the parabola, vertex, and logic trace.

Quadratic Formula Calculator FAQ

What is the quadratic formula?

For ax² + bx + c = 0 (a ≠ 0), the roots are
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The calculator shows roots in simplified radical form and decimal form, and uses i when the discriminant is negative

What does the discriminant tell you?

The discriminant is
D=b2−4acD = b^2 - 4ac
D > 0: two distinct real roots. D = 0: one repeated real root. D < 0: two complex conjugate roots. The tool displays D and the correct root form

How do I get simplified radical form?

When D ≥ 0, the calculator factors perfect squares out of √D (e.g. √20 → 2√5) and writes each root as a real part ± a multiple of √(radicand). Decimal approximations are shown as well.

Why does the parabola not cross the x-axis when D < 0?

When D < 0, the roots are complex, so there are no real x-intercepts. The parabola lies entirely above or below the x-axis; the graph shows the curve and a note that roots are complex.

What is the vertex of a parabola?

For f(x) = ax² + bx + c, the vertex is (h, k) with
h=−b2ah = -\frac{b}{2a}
and
k=f(h)k = f(h)
The axis of symmetry is x = h. The calculator shows the vertex and draws the axis on the parabola

How do I solve a quadratic equation with the calculator?

Enter the coefficients a, b, and c (a ≠ 0). The calculator computes the discriminant and roots, and updates the parabola and Logic Trace. Use Copy results to export the formula, roots, and vertex.

Mathematical Reference Note

Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.

Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.

Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.

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