How this calculator works
The quadratic formula gives the roots: x = (−b ± √(b² − 4ac)) ÷ 2a.
The discriminant b² − 4ac decides the type of roots: positive gives two real roots, zero gives one repeated root, negative gives two complex roots.
The vertex, the parabola’s highest or lowest point, is at x = −b ÷ 2a.
a cannot be 0, because then the equation is linear, not quadratic.
The sum of the roots is always −b ÷ a and their product c ÷ a, a quick way to check an answer. For x² − 3x − 10, the roots 5 and −2 add to 3 and multiply to −10.
Worked example
x² − 3x − 10 = 0
- Discriminant: (−3)² − 4 × 1 × (−10) = 49.
- x = (3 ± 7) ÷ 2, so x = 5 or x = −2.
- Vertex: (1.5, −12.25).
Questions people ask
When should I factor instead?
When the roots are whole numbers, factoring can be quicker: x² − 3x − 10 = (x − 5)(x + 2). The formula always works.
What are complex roots?
When the discriminant is negative there are no real solutions; the roots involve i, the square root of −1.
What does the vertex tell me?
The minimum (if a > 0) or maximum (if a < 0) value of the quadratic, useful in optimisation and physics problems.
Where are quadratics used outside class?
Projectile paths, area and profit problems, and anywhere one quantity depends on the square of another, such as stopping distance at different speeds.
Last reviewed October 2, 2026