Quadratic Formula Calculator
This free quadratic formula calculator solves any quadratic equation ax² + bx + c = 0 in seconds. Enter the coefficients a, b and c to get the discriminant, real or complex roots, the vertex, the y-intercept and the factored form — with every step of the formula shown so you can check your homework.
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Tip: results update instantly as you type. All calculations happen in your browser — your numbers are never sent anywhere. Rounded to at most 6 decimal places.
How to use this calculator
Type the coefficients a, b and c of your equation ax² + bx + c = 0 into the three boxes. The defaults (a = 1, b = −5, c = 6) solve the classic equation x² − 5x + 6 = 0, which factors into (x − 2)(x − 3).
Everything recalculates the moment you change a number — no submit button needed. If you set a = 0, the equation stops being quadratic, so the calculator switches to a linear-equation message instead of dividing by zero. Decimals and negative numbers are welcome.
How it works
The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) works by completing the square on ax² + bx + c = 0, which moves everything except the squared term to one side so x can be isolated. Whatever a, b and c you enter, the same steps apply.
The key quantity is the discriminant D = b² − 4ac, the part under the square root. Its sign reveals the roots before you finish: a positive D gives two distinct real roots (the parabola crosses the x-axis twice), D = 0 gives one repeated root (the vertex touches the x-axis), and a negative D gives two complex roots (the parabola never touches the x-axis). In the complex case the roots come as a conjugate pair p ± qi, where p = −b/(2a) and q = √|D|/(2a).
The vertex at x = −b/(2a) is the parabola's turning point — plug that x back into f(x) = ax² + bx + c to get its height. It is the lowest point when a > 0 and the highest when a < 0, and it always sits exactly halfway between two real roots. The y-intercept is simply c, the point (0, c). When the roots are real, the equation factors as a(x − x₁)(x − x₂), or a(x − x₁)² for a repeated root. Assumptions: inputs are ordinary decimal numbers; results are rounded to at most 6 decimal places. Nothing here is an estimate — every value is exact arithmetic on your coefficients.
Frequently asked questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² − 4ac)) / (2a). It solves any quadratic equation ax² + bx + c = 0, as long as a is not zero — plug in the three coefficients and the formula returns the values of x that make the equation true.
What does the discriminant tell you?
The discriminant D = b² − 4ac reveals the number and type of solutions before you finish solving. A positive D means two distinct real roots; D = 0 means one repeated (double) root; a negative D means two complex roots with an imaginary part.
What happens if a = 0?
If a = 0, the equation is no longer quadratic — it becomes the linear equation bx + c = 0, which has the single solution x = −c / b (as long as b is not zero). This calculator detects that case and shows you the linear solution instead of dividing by zero.
Can this calculator solve quadratics with complex roots?
When the discriminant is negative, the calculator shows the roots in p ± qi form (for example, −1 ± 2i) and walks through the step where the square root of a negative number becomes an imaginary multiple.
What is the vertex, and why does it matter?
The vertex is the parabola's turning point, at x = −b / (2a) with y = f(−b / (2a)). It is the minimum when a is positive (parabola opens up) and the maximum when a is negative. You can sanity-check your roots against it: the vertex always sits exactly halfway between two real roots.