Math calculator
Quadratic Formula Calculator
Solve ax² + bx + c = 0 with the quadratic formula: real or complex roots, discriminant, vertex.
Quick answer
How do you use the quadratic formula?
Write the equation as ax² + bx + c = 0, then plug a, b, and c into x = (−b ± √(b² − 4ac)) ÷ 2a. First find the discriminant b² − 4ac: if it is positive there are two real roots, if it is 0 there is one, and if it is negative the roots are complex.
x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant Δ = b² − 4ac. Vertex x = −b ÷ 2a.
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- Equation: 1x² − 5x + 6 = 0, so a = 1, b = -5, c = 6.
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
- Discriminant: b² − 4ac = 25 − 24 = 1.
- The discriminant is positive, so there are two real roots.
- x = (5 ± √1) ÷ 2 → x = 2 or x = 3
- Vertex: x = −b ÷ 2a = 2.5, y = -0.25. The parabola opens upward and crosses the y-axis at 6.
Enter a, b, and c from ax² + bx + c = 0. The calculator finds the roots with the quadratic formula, including complex answers, and shows the discriminant, vertex, and every step.
About this Quadratic Formula Calculator
The Quadratic Formula Calculator solves any equation in the form ax² + bx + c = 0. Enter a, b, and c to get the roots — real or complex — plus the discriminant, vertex, and a step-by-step solution you can follow for homework.
What you'll need
- • a — the number in front of x² (cannot be 0)
- • b — the number in front of x
- • c — the constant term
How it works
- 1. Rearrange the equation so one side is 0: ax² + bx + c = 0.
- 2. Identify a, b, and c, keeping their signs.
- 3. Compute the discriminant b² − 4ac.
- 4. Plug into x = (−b ± √Δ) ÷ 2a and simplify both the + and − versions.
- 5. Check your answers by substituting them back into the equation.
Worked example
For x² − 5x + 6 = 0: a = 1, b = −5, c = 6. Discriminant = 25 − 24 = 1. x = (5 ± 1) ÷ 2, so x = 3 or x = 2. The vertex is at (2.5, −0.25).
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Quadratic Formula Calculator FAQ
What is the Quadratic Formula Calculator?
The Quadratic Formula Calculator is a free online tool that helps you solve ax² + bx + c = 0 with the quadratic formula: real or complex roots, discriminant, vertex.
How do I use the Quadratic Formula Calculator?
Open the Quadratic Formula Calculator, enter the requested values, and review the instant estimate on the page.
Is the Quadratic Formula Calculator free?
Yes. The Quadratic Formula Calculator is free to use on Fast Online Calculator and does not require an account.
What does the discriminant tell you?
The discriminant b² − 4ac tells you how many solutions there are: positive means two real roots, zero means one repeated root, and negative means two complex roots.
Can the quadratic formula give complex answers?
Yes. When b² − 4ac is negative, the square root of a negative number appears, so the roots are written as p ± qi, where i = √−1.
What if a is 0?
If a = 0 the equation is linear (bx + c = 0), not quadratic. Solve it as x = −c ÷ b, or use the Algebra Solver.
How do I find the vertex of a parabola?
The vertex x-coordinate is −b ÷ 2a. Plug that x back into ax² + bx + c to get the y-coordinate.
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