Quadratic Formula Calculator
Roots, vertex and discriminant for ax² + bx + c.
Enter your values
Enter the a you are working with.
Enter the b you are working with.
Enter the c you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Roots
- x = 3, x = 2
- Discriminant
- 1 — two real roots
- Vertex
- (2.5, -0.25)
- Axis of symmetry
- x = 2.5
Working
- 1Discriminant = -5² − 4(1)(6) = 1
Quick answer
The quadratic formula gives x = (−b ± √(b² − 4ac)) ÷ 2a; the discriminant b² − 4ac tells you how many real roots exist.
x = (−b ± √(b² − 4ac)) ÷ 2a
At a glance
| What it does | Roots, vertex and discriminant for ax² + bx + c. |
|---|---|
| Category | Math |
| Inputs needed | a, b, c |
| Main output | Roots |
| Formula | x = (−b ± √(b² − 4ac)) ÷ 2a |
| Cost | Free — no sign-up, no download |
How to use the Quadratic Formula Calculator
- 1Enter the coefficients a, b and c.
- 2Read the roots and discriminant.
- 3Use the vertex to sketch the parabola.
Inputs explained
- a
- Enter the a you are working with.
- b
- Enter the b you are working with.
- c
- Enter the c you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- a1
- b-5
- c6
Results
- Rootsx = 3, x = 2
- Discriminant1 — two real roots
- Vertex(2.5, -0.25)
- Axis of symmetryx = 2.5
- Discriminant = -5² − 4(1)(6) = 1
Frequently asked questions
What does the discriminant tell me?
Positive means two real roots, zero means one repeated root, negative means the parabola never crosses the x-axis.
Can I always factor instead?
Only when roots are rational. The formula works for every quadratic, factorable or not.
What do I need to enter into the Quadratic Formula Calculator?
Just 3 values: a, b, c. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Quadratic Formula Calculator work out the answer?
The quadratic formula gives x = (−b ± √(b² − 4ac)) ÷ 2a; the discriminant b² − 4ac tells you how many real roots exist. It applies the formula x = (−b ± √(b² − 4ac)) ÷ 2a and shows the working so you can check each step by hand.
Is the Quadratic Formula Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.