Quadratic Formula Calculator
Solve ax² + bx + c = 0 for x, including cases with complex roots.
How this calculator works
The quadratic formula solves any equation of the form ax² + bx + c = 0. When the discriminant is negative, the roots are complex numbers.
a, b, c are the coefficients of the quadratic equation.
Understanding the discriminant
The discriminant (b² − 4ac) tells you what type of roots to expect before even solving the full equation. A positive discriminant means two distinct real roots, zero means exactly one repeated real root, and a negative discriminant means two complex (non-real) roots.
This is useful for quickly checking whether a quadratic equation has real solutions at all — in applied contexts (like physics problems), a negative discriminant often signals that the scenario as posed has no real physical solution.
A worked example
For x squared minus 5x plus 6 equals 0 (a=1, b=-5, c=6): the discriminant is 25 minus 24, or 1 (positive), giving two real roots: x equals (5 plus or minus 1) divided by 2, so x equals 3 or x equals 2.
For x squared plus 2x plus 5 equals 0 (a=1, b=2, c=5): the discriminant is 4 minus 20, or negative 16, giving complex roots: x equals negative 1 plus or minus 2i, since there's no real number solution to this equation.
Interpreting the discriminant
The discriminant (b² − 4ac) reveals the nature of a quadratic equation's roots before you even solve for them: a positive discriminant means two distinct real roots, zero means exactly one repeated real root (the parabola touches the x-axis at a single point), and a negative discriminant means two complex roots (the parabola never crosses the x-axis).
This connects directly to graphing — a quadratic equation's roots are the x-intercepts of its parabola, so the discriminant essentially tells you how many times (if any) the parabola crosses the x-axis.
Frequently asked questions
What if the discriminant is negative?
The equation has two complex (non-real) roots. This calculator shows them in the form a ± bi.
What if a = 0?
The equation is no longer quadratic (it becomes linear), so the quadratic formula doesn't apply — solve bx + c = 0 directly instead.
What does it mean if the roots are complex numbers?
It means the equation has no real number solutions — graphically, the parabola never crosses the x-axis. Complex roots are still mathematically valid but represent no real x-intercept.
Can I use this for equations not already in ax² + bx + c = 0 form?
Rearrange your equation into that standard form first — move all terms to one side and set equal to zero, then identify your a, b, and c coefficients.
What does it mean if the discriminant is zero?
It means the quadratic has exactly one repeated real root — graphically, the parabola's vertex touches the x-axis at exactly one point rather than crossing it.