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Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 with the discriminant, exact roots in radical form, complex roots, vertex form, factored form and a graph.

Equation ax² + bx + c = 0

Integers, decimals and fractions such as 1/2 are accepted. An empty b or c counts as 0.

Examples

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    Step-by-step solution

    How to use the quadratic equation solver

    1. Type the three coefficients of ax² + bx + c = 0 in Coefficient a, Coefficient b and Coefficient c. Integers, decimals and fractions such as 1/2 are accepted, and an empty b or c counts as 0.
    2. The answer updates as you type; Solve does the same. Clear empties the three boxes and Copy copies the equation and its solutions.
    3. Read the Solutions: each root is shown in exact form, followed by a decimal approximation when the exact value is not a short decimal.
    4. Below, the tool lists the Discriminant Δ, the Vertex, the Axis of symmetry, the y-intercept, the direction in which the parabola opens and the sum and product of the roots. The three Forms of the function (standard, vertex and factored) and a Graph of the parabola follow.
    5. The Step-by-step solution repeats the whole working: discriminant, square root of Δ, quadratic formula, both roots, vertex, factored form and a check.

    The Examples buttons load typical cases: two whole-number roots, a double root, complex roots and irrational roots.

    The quadratic formula and the discriminant

    A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its solutions are given by the quadratic formula:

    x = (−b ± √Δ) / (2a), where Δ = b² − 4ac

    The discriminant Δ decides everything:

    • Δ > 0: two distinct real roots. If Δ is a perfect square (and the coefficients are rational), the roots are rational and the polynomial factors neatly, for example x² − 5x + 6 = (x − 2)(x − 3).
    • Δ = 0: one double root, x = −b / (2a). The parabola touches the x-axis at its vertex.
    • Δ < 0: no real root. The square root of a negative number is imaginary, √Δ = i√|Δ|, so the roots are two complex conjugates such as −1 ± 2i.

    This tool computes with exact fractions, so it never loses precision. When √Δ is not a whole number, it simplifies the radical (√32 = 4√2) and reduces the result: x² + 3x + 1 = 0 gives x = (−3 ± √5)/2, and x² − 8 = 0 gives x = ±2√2.

    Vertex form, factored form and the graph

    The vertex form a(x − h)² + k shows the vertex (h, k) directly, with h = −b / (2a) and k = −Δ / (4a). The line x = h is the axis of symmetry: both roots are the same distance from it. When a > 0 the parabola opens upward and k is its minimum; when a < 0 it opens downward and k is its maximum.

    The factored form a(x − x₁)(x − x₂) uses the roots. With rational roots the tool writes integer factors, for example 2x² − x − 1 = (2x + 1)(x − 1). When Δ < 0 it cannot be factored over the real numbers, and the tool says so before showing the complex factors.

    The graph is drawn to scale around the vertex and the roots. Blue points mark the real roots, the orange point marks the vertex and the dashed line is the axis of symmetry.

    As a final check the tool applies Vieta’s formulas: x₁ + x₂ = −b/a and x₁ × x₂ = c/a.

    Frequently asked questions

    What happens if a = 0?

    The equation is then linear, bx + c = 0, and the tool solves it as such: x = −c / b. If b is also 0, either every number is a solution (c = 0) or there is none (c ≠ 0).

    Why do I get a result with i?

    The letter i is the imaginary unit, with i² = −1. It appears when the discriminant is negative: the parabola never crosses the x-axis, so there is no real solution, but there are two complex ones.

    Can I enter decimals or fractions as coefficients?

    Yes. Type 0.5, 1/3 or −2.25; the calculation stays exact because every coefficient is converted to a fraction first.

    How accurate are the decimal values?

    The exact form is always correct. Decimal approximations are rounded to seven significant digits, which is enough for homework and most practical uses.

    How do I find the vertex of a parabola?

    Use h = −b / (2a), then k = f(h), which also equals −Δ / (4a). The tool shows both formulas with your numbers in the step-by-step solution.

    Is the double root counted once or twice?

    As a solution of the equation it is a single value, shown once. As a root of the polynomial it has multiplicity two, which is why the factored form contains a square, such as 2(x − 1)².

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