The formula
How it works
The quadratic formula solves any equation of the form ax² + bx + c = 0. Enter the three coefficients and the calculator returns both roots — real or complex — along with the discriminant that tells you which kind to expect.
FAQ
What does the discriminant tell me?
The discriminant is b² − 4ac. If it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the two roots are complex (they involve i).
Why must a not be zero?
If a is zero the equation is no longer quadratic — it becomes linear, bx + c = 0, with a single solution. The quadratic formula divides by 2a, so a must be non-zero.
How is the quadratic formula different from factoring?
Factoring finds the same roots by rewriting ax² + bx + c as a product of two simpler terms, but it only works neatly when the roots are rational. The quadratic formula always works, even when the roots are irrational or complex.
What do complex roots mean physically?
Complex roots mean the parabola never crosses the x-axis, so the equation has no real-number solution. In applied problems this often signals that the situation you are modelling (like a trajectory hitting a target height) simply cannot happen with those inputs.
Can I check the roots without recomputing everything?
Yes — Vieta’s formulas say the sum of the roots equals −b⁄a and their product equals c⁄a. Adding and multiplying the two roots the calculator gives you is a quick sanity check.
How do the roots relate to the parabola’s graph?
The roots are the x-values where the graph of y = ax² + bx + c crosses the x-axis. The vertex sits exactly halfway between two real roots, at x = −b⁄2a.
Does the sign of a matter for the shape of the graph?
Yes. A positive a makes the parabola open upward with a minimum point, while a negative a makes it open downward with a maximum point — but the quadratic formula finds the roots the same way in both cases.
About the quadratic formula calculator
This calculator solves quadratic equations — those that can be written as ax² + bx + c = 0 — using the quadratic formula. Quadratics appear whenever something depends on a squared term, from the path of a thrown ball to the area of a shape, and finding where they equal zero is one of the most common tasks in algebra. The calculator does the arithmetic exactly, including the awkward cases where the answers are not whole numbers or not even real.
How to use it
Enter the three coefficients a, b and c from your equation, keeping the signs. The calculator shows the discriminant, then both roots. For example, x² − 3x + 2 = 0 has coefficients 1, −3 and 2, and the calculator returns the roots x = 2 and x = 1. If the discriminant is negative, the roots are shown as complex numbers with a real and an imaginary part. Make sure a is not zero, or the equation is not a quadratic.
The formula
The quadratic formula is . The part under the square root, , is the discriminant, and its sign decides everything: positive gives two real roots, zero gives one, and negative gives a complex pair. The means you compute two answers — one adding the root and one subtracting it. It is derived by completing the square on the general equation.
Where it is used
Quadratics model any situation with a squared relationship. Physics uses them for projectile motion and free fall, where height depends on time squared; engineering uses them for areas, trajectories and optimisation; and finance uses them in some interest and break-even problems. Being able to solve them quickly is a foundation for higher maths, and the discriminant alone is often enough to tell whether a real solution exists at all.