Free Maths Tool

Quadratic Equation Solver

Enter a, b and c to solve any quadratic equation in seconds. This quadratic equation solver gives you exact roots, the discriminant, the vertex, a graph of the parabola and the full working, so you can check your answer and understand each line.

  • Exact and decimal roots
  • Complex solutions included
  • Step by step working
  • Graph drawn instantly
Your equation
x² + x + = 0
Try an example

Type your coefficients and the answer updates instantly.

Show the full working
    Two real roots
    • 1Discriminant
    • (0, 0)Vertex
    • x = 0Axis of symmetry
    • UpwardParabola opens
    Graph of the parabola

    Why this quadratic solver is more useful

    Plenty of sites return two decimals and stop there. If you are learning, that is the least useful part of the answer.

    Exact surd answers

    Roots are simplified into exact form such as (3 plus root 5) over 2, not just a rounded decimal, which is what most exam papers ask for.

    Free step by step working

    Every substitution is shown, from the discriminant to the final simplification. No paywall and no account to see the method.

    Graph drawn instantly

    The parabola is plotted with its roots and vertex marked, so you can see whether the curve crosses the x axis once, twice or never.

    Complex roots supported

    When the discriminant is negative you still get an answer, written in proper a plus bi form rather than an error message.

    Vertex and axis included

    The turning point, axis of symmetry and direction of opening come with every answer, which covers most follow up questions in one go.

    No ads, no waiting

    Results update as you type, the page never reloads, and nothing you enter is sent to a server.

    How to solve a quadratic equation here

    1. Write the equation in standard form

      Rearrange it so everything sits on one side as ax squared plus bx plus c equals zero. Move terms across before you read off the coefficients.

    2. Enter a, b and c

      Type each coefficient into its box, using a minus sign where needed. Leave a term as zero if it is missing from your equation.

    3. Read the roots and the discriminant

      The discriminant tells you what kind of answer to expect before you even look at the roots, and the graph confirms it visually.

    4. Open the working to check your method

      Expand the working panel to see each substitution, then compare it line by line with what you wrote on paper.

    The quadratic formula and the discriminant

    Every quadratic in the form ax squared plus bx plus c equals zero is solved by the same formula: x equals minus b, plus or minus the square root of b squared minus four ac, all divided by two a. The part under the root sign is the discriminant, and it decides the shape of the answer before you calculate anything else.

    DiscriminantWhat you getOn the graph
    Greater than zeroTwo different real rootsThe parabola crosses the x axis twice
    Exactly zeroOne repeated real rootThe parabola just touches the x axis at the vertex
    Less than zeroTwo complex roots, a pair of conjugatesThe parabola never meets the x axis
    A perfect squareTwo rational roots, so the equation factorises neatlyBoth crossings land on tidy values

    The vertex sits at x equals minus b over two a, which is also the axis of symmetry. If a is positive the parabola opens upward and the vertex is the minimum point. If a is negative it opens downward and the vertex is the maximum.

    Common mistakes worth avoiding

    • Forgetting to rearrange first. The formula only works once the equation equals zero, so move every term across before reading off a, b and c.
    • Dropping a minus sign. If b is negative then minus b is positive. This single slip causes more wrong answers than anything else in the topic.
    • Squaring b incorrectly. When b is negative, b squared is still positive. Minus four squared is sixteen, not minus sixteen.
    • Dividing only part of the numerator. The whole of minus b plus or minus the root is divided by two a, not just the root.
    • Rounding too early. Keep the surd until the final line, then round once. Rounding mid calculation shifts the answer.
    • Assuming no solution. A negative discriminant does not mean there is no answer, only that the two roots are complex.
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    Quadratic equation solver FAQ

    What is a quadratic equation?
    It is any equation that can be written as ax squared plus bx plus c equals zero, where a is not zero. The highest power of x is two, which is what makes the graph a parabola rather than a straight line.
    Why must a not be zero?
    If a is zero the x squared term disappears and you are left with a linear equation, bx plus c equals zero. The quadratic formula also divides by two a, so a zero value would make it undefined. Solve linear equations by rearranging instead.
    What does the discriminant tell me?
    It is b squared minus four ac, the part under the square root. A positive value gives two real roots, zero gives one repeated root, and a negative value gives a pair of complex roots. If it is a perfect square, the equation factorises into whole numbers.
    Can this solver handle complex roots?
    Yes. When the discriminant is negative the answer is shown as a real part plus or minus an imaginary part, in the standard a plus bi form, with the surd simplified rather than left as a rounded decimal.
    Should I factorise or use the formula?
    Try factorising first when the numbers are small and the discriminant is a perfect square, because it is faster. Reach for the formula whenever the coefficients are awkward, the roots are irrational, or you cannot spot the factors within about thirty seconds.
    How do I find the vertex of a parabola?
    The x coordinate of the vertex is minus b divided by two a. Substitute that back into the equation to get the y coordinate. This tool does both automatically and marks the point on the graph.
    Can I use decimals or fractions as coefficients?
    Decimals work directly. For fractions, multiply the whole equation through by the common denominator first, which clears them without changing the roots and keeps the working much tidier.
    Where are quadratics used outside school?
    They model anything with a squared relationship, including projectile paths, braking distances, profit against price, satellite dish shapes and areas of rectangular land. Finding the vertex is often the point, since it gives the maximum or minimum.

    Check your answer in seconds

    Bookmark this quadratic equation solver for homework, revision and exam practice. It is free, it shows every step, and it works on any device.

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