Quadratic formula calculator
Solve ax² + bx + c = 0 for real or complex roots, and see the parabola, vertex and discriminant.
1x² + (-3)x + (-4) = 0
Gold dots are the real roots (x-intercepts), the red dot is the vertex, the dashed line is the axis of symmetry.
The formula
x = (−b ± √(b² − 4ac)) / 2a
It comes from completing the square on ax² + bx + c = 0. The −b/2a part is the vertex’s x-coordinate; the ± √ part is how far each root sits either side of it.
Show me the steps
An AI tutor walks through the working for this exact result. The answer above is computed in your browser; the explanation is optional and is the only thing that leaves your device.
Solve 1x² + (-3)x + (-4) = 0 using the quadratic formula = x₁ = 4, x₂ = −1
Questions people ask
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0: x = (−b ± √(b² − 4ac)) / (2a). The ± gives the two roots.
What does the discriminant tell me?
b² − 4ac decides the roots before you finish: positive means two real roots (the parabola crosses the x-axis twice), zero means one repeated root (it touches the axis), negative means two complex roots (it never reaches the axis).
Can the calculator give complex roots?
Yes. When the discriminant is negative, the roots are shown as p ± qi, where i = √−1.
When should I factor instead?
If the roots are whole numbers, factoring is faster: x² − 3x − 4 = (x − 4)(x + 1). The formula always works, including when factoring is impossible over the integers.