Quadratics describe projectile arcs, profit curves, areas, stopping distances: any time a rate itself changes steadily, x² appears. Solving them by formula is mechanical once you have the pieces.
Updated August 15, 2026, after a fresh review, with updated visuals and links.
Enter a, b and c above and get the discriminant verdict plus the roots (real or complex) instantly. The guide below explains what the discriminant is really telling you.
The discriminant decides everything
D = b squared minus 4ac is the oracle: positive D means the parabola crosses the x-axis twice (two real roots), zero D means it kisses the axis once (a repeated root), negative D means it never touches (two complex roots).
Worked example: x squared minus 5x plus 6 has D equal to 25 minus 24, which is 1. Roots are (5 plus 1)/2 and (5 minus 1)/2: exactly 3 and 2.
Physics flavor: a ball launched upward with height h(t) = minus 4.9t² + 20t + 2 asks 'when does it land?' Setting the height to zero and solving gives you the flight time, the positive root.
Worth remembering
- Sanity check by substitution: plug each root back into the original equation; it should evaluate to zero within rounding.
- Pausing on negative D: in engineering contexts, complex roots point at oscillations with decay, not a mistake in your setup.
Frequently asked
What is the quadratic formula?
x equals negative b, plus or minus the square root of (b squared minus 4ac), all over 2a. The tool applies it and shows the discriminant along the way.
What does a negative discriminant mean?
The parabola never crosses the x-axis, so there are no real number solutions: the two roots are complex conjugates, which the tool displays in a plus-minus-bi form.
