Everything is worked out in your browser and saved on this device for next time.
About the quadratic formula calculator
A quadratic equation has the form ax² + bx + c = 0, with a not zero. Its solutions (also called roots) come from the quadratic formula:
x = (−b ± √(b² − 4ac)) / (2a)
Type a, b and c. The calculator works through the formula step by step: the discriminant, the square root, then both roots. It writes the roots as exact fractions or roots when it can (x = 1 ± √2), finds complex roots when there are no real ones, and draws the parabola with its roots and vertex.
Features
- Every step — the discriminant, the square root and both roots, with your numbers
- Exact form — fractions (1/2), simplified roots (1 ± √2) and complex roots (−1 ± 2i)
- Real or complex — the discriminant tells which, and the calculator says so
- Factored form — like (2x − 1)(x + 3), when the roots are fractions
- Vertex and vertex form — the turning point, the axis of symmetry and y = a(x − h)² + k
- A graph — the parabola with its roots, vertex and axis of symmetry
- Decimals and fractions welcome — type 0.5, 1/2 or −3
- Saved on your device — your equation is still there next time
What the discriminant tells you
The discriminant is D = b² − 4ac, the part under the square root.
| D | Roots | The graph |
|---|---|---|
| D > 0 | two different real roots | crosses the x-axis twice |
| D = 0 | one double root | touches the x-axis once |
| D < 0 | no real roots, two complex roots | does not reach the x-axis |
If D is a perfect square (0, 1, 4, 9, 16 …) and a, b, c are whole numbers, the roots are fractions and the equation can be factored.
Worked examples
Two real roots: 2x² + 5x − 3 = 0. a = 2, b = 5, c = −3.
D = 5² − 4 × 2 × (−3) = 25 + 24 = 49. √49 = 7.
x = (−5 ± 7) / 4, so x₁ = 2/4 = 1/2 and x₂ = −12/4 = −3. Factored: (2x − 1)(x + 3).
Roots with a square root: x² − 2x − 1 = 0.
D = 4 + 4 = 8, and √8 = 2√2. x = (2 ± 2√2) / 2 = 1 ± √2, so x₁ ≈ 2.4142 and x₂ ≈ −0.4142.
One double root: x² − 4x + 4 = 0.
D = 16 − 16 = 0, so x = 4 / 2 = 2. Factored: (x − 2)².
Complex roots: x² + 2x + 5 = 0.
D = 4 − 20 = −16, and √(−16) = 4i. x = (−2 ± 4i) / 2 = −1 ± 2i. The parabola stays above the x-axis; its vertex is (−1, 4).
The vertex
The vertex is the turning point of the parabola: h = −b / (2a) and k = c − b² / (4a). For 2x² + 5x − 3 the vertex is (−5/4, −49/8) = (−1.25, −6.125). The line x = h is the axis of symmetry: the two roots sit at the same distance on each side of it.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / (2a). It solves every equation of the form ax² + bx + c = 0, as long as a is not 0.
What if a = 0?
Then it is not a quadratic. The equation is bx + c = 0, a straight line, and x = −c ÷ b. The calculator shows that answer and says so.
What are complex roots?
When the discriminant is negative, there is no real number whose square is negative, so the roots use i, where i² = −1. They always come in pairs like −1 + 2i and −1 − 2i.
Can I type fractions or decimals?
Yes. Type 1/2, 0.5 or −3. Decimals are turned into whole numbers inside (0.5x² − 1.5x + 1 becomes x² − 3x + 2), so the exact form still works.
Why is there no factored form?
The factored form with whole numbers only exists when the roots are fractions. x² − 2x − 1 has roots 1 ± √2, so it does not factor with whole numbers.
Related tools
- Graphing Calculator — plot any function and find its roots
- Fraction Calculator — add, subtract and simplify fractions
- Pythagorean Theorem Calculator — the third side of a right triangle
- GCF and LCM Calculator — common factors, used to simplify roots
- Scientific Calculator — powers and square roots
How to use it
- Type a, b and c — the equation appears above the boxes
- Use a minus sign for negative numbers, and 0 for a missing term
- Read the roots in the result box, with the exact form, the discriminant and the vertex
- Follow the steps under the form and look at the graph
- Or tap an example to see each kind of answer
