Quadratic Formula Calculator
Enter a, b and c for ax² + bx + c = 0 to get both roots exactly - as whole numbers, fractions, simplified square roots or complex numbers - plus decimals, the discriminant, the vertex and the factored form, with each step of the quadratic formula shown.
x² +
x +
= 0
Use whole numbers, decimals (0.5) or fractions (1/2). For x² − 3 = 0 enter a = 1, b = 0, c = −3.
Roots
-
-Discriminant b² − 4ac
-Vertex
-Axis of symmetry
-Factored form
Steps
Three ways to solve a quadratic
| Method | When it works | Example |
|---|---|---|
| Factoring | When the roots are rational and easy to spot | x² − 5x + 6 = (x − 2)(x − 3) → x = 2, 3 |
| Completing the square | Always; useful for finding the vertex form | x² − 6x + 5 = (x − 3)² − 4 → vertex (3, −4) |
| Quadratic formula | Always, including irrational and complex roots | x = (−b ± √(b² − 4ac)) ÷ 2a |
This calculator uses the formula, because it works for every quadratic, and gives the factored form whenever the roots are rational - so you can check a factoring answer too.
Worked example: 2x² + 4x − 1 = 0
- Identify a, b and c
a = 2, b = 4, c = −1 - Discriminant
D = 4² − 4 × 2 × (−1) = 16 + 8 = 24 - Simplify the square root
√24 = √(4 × 6) = 2√6 - Apply the formula and reduce by 2
x = (−4 ± 2√6) ÷ 4 = (−2 ± √6) / 2 - Decimals
x ≈ 0.2247 and x ≈ −2.2247
Reading the discriminant
Before solving, D = b² − 4ac tells you what kind of answer to expect:
- D > 0 and a perfect square: two rational roots, and the quadratic factors neatly.
- D > 0, not a perfect square: two irrational roots, written with a simplified square root.
- D = 0: one repeated root; the parabola just touches the x-axis at its vertex.
- D < 0: no real roots; the parabola never meets the x-axis, and the roots are a pair of complex numbers p ± qi.
The sign of a decides which way the parabola opens: up when a is positive, down when it is negative. The vertex is the lowest or highest point.
What this calculator does not do
- Quadratics only: it does not solve cubic or higher-degree equations.
- Coefficients must be real numbers; complex coefficients are not supported.
- Square roots are simplified by checking factors up to 100,000; with very large coefficients the radical may not be fully simplified, though it is still exact.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a not zero, x = (−b ± √(b² − 4ac)) ÷ 2a. The ± gives the two roots: one with plus, one with minus.
What does the discriminant tell me?
The discriminant is D = b² − 4ac. If D is positive there are two different real roots, if D is 0 there is one repeated real root, and if D is negative there are two complex roots and the parabola never crosses the x-axis.
Why are some answers written with a square root?
When D is not a perfect square the roots are irrational, so any decimal is only an approximation. The exact answer keeps the square root in its simplest form - √12 becomes 2√3 - and the decimal is shown next to it.
What happens if a is 0?
Then the x² term disappears and the equation is linear, bx + c = 0. The calculator says so and solves it as x = −c ÷ b instead of using the quadratic formula, which would divide by zero.
Can I enter fractions or decimals?
Yes. Enter 1/2 or 0.5. The calculator multiplies the whole equation by a common denominator first, which keeps every step exact and does not change the roots.
How do I find the vertex?
The vertex of y = ax² + bx + c is at x = −b ÷ 2a, and its y-value is c − b² ÷ 4a. For x² − 6x + 5 that is (3, −4).
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