Solve quadratic equations of the form ax² + bx + c = 0. Calculate real and complex roots, vertex coordinates, and plot key characteristics.

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    How to Use This Calculator

    Understanding Quadratic Equations

    A quadratic equation is a second-order polynomial equation in a single variable. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upward or downward.

    Parabola Features

    1. The Discriminant ($D = b^2 - 4ac$):
      • If $D > 0$: The parabola crosses the x-axis at two distinct points (two real roots).
      • If $D = 0$: The parabola touches the x-axis at exactly one point (one real repeated root).
      • If $D < 0$: The parabola does not touch the x-axis, and its roots are complex conjugate pairs.
    2. The Vertex ($h, k$): The peak or trough of the parabola. It represents the maximum or minimum value of the quadratic function.
    3. Leading Coefficient ($a$): Determines the width of the parabola and its direction. If $a$ is positive, it opens upward. If $a$ is negative, it opens downward.

    The Math Behind It

    The tool uses the following mathematical principle:

    $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

    Where:

    • $x$ is the unknown variable we are solving for.
    • $a$ is the quadratic coefficient ($a eq 0$).
    • $b$ is the linear coefficient.
    • $c$ is the constant term.