Quadratic formula calculator
Solve ax² + bx + c = 0 with exact roots, the discriminant, vertex and factored form.
- 🇺🇸 US federal & state rules
Two real roots
x = 3, 2
x² − 5x + 6 = 0
- Discriminant
- 1 (two real roots)
- Decimal roots
- 3, 2
- Vertex
- (5/2, −1/4)
- Factored form
- (x − 3)(x − 2) = 0
This is an example. Enter your numbers above and press the button to see your own result.
Graph and root facts
| Opens | Upward (a > 0), vertex is a minimum |
|---|---|
| Axis of symmetry | x = 5/2 |
| y-intercept | (0, 6) |
| Sum of roots (−b/a) | 5 |
| Product of roots (c/a) | 6 |
Show the math
| Equation | x² − 5x + 6 = 0 |
|---|---|
| Discriminant b² - 4ac | (−5)² − 4 × 1 × 6 = 1 |
| Quadratic formula | x = (5 ± √1) / 2 |
| D > 0, two real roots | x = 3, 2 |
What to do next
How it works
The coefficients are read as exact fractions and scaled to whole numbers. The discriminant b² - 4ac tells you which kind of roots you get, and the square root is simplified, so √8 becomes 2√2.
When the roots are rational, the calculator also writes the factored form. It shows the vertex (-b/2a, f(-b/2a)), the axis of symmetry, and the sum (-b/a) and product (c/a) of the roots.
Things to keep in mind
Move every term to one side first so the equation reads ax² + bx + c = 0. For 2x² = 3x + 5, enter a = 2, b = -3, c = -5.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0, x = (-b ± √(b² - 4ac)) / 2a. For x² - 5x + 6 = 0 that gives (5 ± 1) / 2, so x = 3 or x = 2.
What does the discriminant tell you?
b² - 4ac decides the roots. Positive means two real roots, 0 means one repeated root, and negative means two complex roots with an i.
How do I find the vertex of a parabola?
The x value is -b / 2a and the y value is the function at that x. For x² - 5x + 6 the vertex is (5/2, -1/4).
Why is my answer written with a square root?
When b² - 4ac is not a perfect square, the exact roots are irrational. x² - 2x - 1 = 0 has roots 1 ± √2, about 2.414 and -0.414.