Last updated 28 August 2026
What the discriminant decides
The part under the square root in the quadratic formula, b² − 4ac, determines everything about the answer before you finish calculating it. If it is positive the square root is a real number and the parabola crosses the x-axis twice. If it is exactly zero the two roots collapse into one and the curve touches the axis at a single point — the vertex. If it is negative the square root is imaginary, the parabola never reaches the axis, and the roots come as a conjugate pair. The vertex is independent of all this: it always sits at x = −b ÷ 2a, which is also the axis of symmetry, because a parabola is symmetric about the line through its turning point.
Common questions
What if a is zero?
Then it is not a quadratic at all — the x² term disappears and you are left with a linear equation, bx + c = 0. The solver handles that case separately and gives the single root rather than failing.
What does a negative discriminant mean?
The equation has no real solutions — the parabola never crosses the x-axis. The roots still exist as complex numbers in the form p ± qi, and both are shown, because they are perfectly valid answers rather than an error.
How is the vertex worked out?
The x coordinate is −b ÷ 2a, which is also the axis of symmetry. Substituting that back into the equation gives the y coordinate. It is the turning point of the curve — the minimum if a is positive, the maximum if a is negative.
Why is the factored form sometimes not shown?
Factored form is only written out when the roots are real. With complex roots the factorisation involves imaginary numbers, which is more confusing than helpful in this context, so the roots themselves are shown instead.