Quadratic Formula Calculator (2026)
Solving ax² + bx + c = 0 by hand means carefully tracking signs under a square root — one small slip and the answer is wrong. Enter your a, b, and c coefficients below and this calculator instantly solves for x using the quadratic formula, correctly handling real roots, a repeated root, or complex roots when the discriminant is negative — with every step of the working shown, plus the discriminant, vertex, and axis of symmetry.
x = (−b ± √(b² − 4ac)) / (2a)
The value under the square root, b² − 4ac (the discriminant), tells you what kind of roots you'll get: two real roots (positive), one repeated real root (zero), or two complex roots (negative).
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The defining feature is the x² term — the highest power of the variable is exactly 2. Graphed on a coordinate plane, every quadratic equation forms a parabola, a symmetric U-shaped (or upside-down U) curve.
- Every quadratic equation has exactly two solutions (roots), counting repeats and complex values
- The roots are the x-values where the parabola crosses (or touches) the x-axis
- If a > 0, the parabola opens upward; if a < 0, it opens downward
- The quadratic formula solves every quadratic equation, even ones that don't factor neatly
- Quadratics model projectile motion, area problems, and many optimization problems
The Quadratic Formula
This formula is derived by completing the square on the general equation ax² + bx + c = 0, and it works for absolutely any valid quadratic equation — regardless of whether the roots are whole numbers, fractions, irrational numbers, or complex numbers. The "±" symbol indicates there are generally two solutions: one using +√ and one using −√.
The Discriminant Explained
The expression under the square root, b² − 4ac, is called the discriminant. Its sign alone tells you what kind of roots the equation has, before you even finish solving:
| Discriminant Value | Root Type | Graph Behavior |
|---|---|---|
| b² − 4ac > 0 | Two distinct real roots | Parabola crosses the x-axis at 2 points |
| b² − 4ac = 0 | One repeated real root | Parabola touches the x-axis at exactly 1 point (its vertex) |
| b² − 4ac < 0 | Two complex conjugate roots | Parabola never touches the x-axis |
How to Use the Quadratic Formula (Step by Step)
- Step 1: Write the equation in standard form ax² + bx + c = 0, identifying a, b, and c.
- Step 2: Calculate the discriminant: b² − 4ac.
- Step 3: If the discriminant is negative, note that roots will be complex; otherwise take its square root.
- Step 4: Substitute a, b, and the square root of the discriminant into x = (−b ± √(b² − 4ac)) / (2a).
- Step 5: Calculate the "+" version and the "−" version separately to get both roots.
- Step 6: Simplify each root — reduce fractions or simplify radicals where possible.
Worked Examples
x² − 3x + 2 = 0 → a=1, b=−3, c=2
Discriminant = (−3)² − 4(1)(2) = 9 − 8 = 1
x = (3 ± √1) / 2 → x = (3+1)/2 = 2, or x = (3−1)/2 = 1
x² − 4x + 4 = 0 → a=1, b=−4, c=4
Discriminant = (−4)² − 4(1)(4) = 16 − 16 = 0
x = (4 ± 0) / 2 = 2 (repeated root)
x² + 2x + 5 = 0 → a=1, b=2, c=5
Discriminant = (2)² − 4(1)(5) = 4 − 20 = −16
x = (−2 ± √−16) / 2 = (−2 ± 4i) / 2 = −1 + 2i or −1 − 2i
2x² + 5x − 3 = 0 → a=2, b=5, c=−3
Discriminant = 25 − 4(2)(−3) = 25 + 24 = 49
x = (−5 ± 7) / 4 → x = 2/4 = 0.5, or x = −12/4 = −3
Other Ways to Solve Quadratics
| Method | Best When | Limitation |
|---|---|---|
| Factoring | Roots are simple rational numbers | Doesn't work for irrational/complex roots |
| Completing the square | You need the vertex form as well as roots | More steps than the formula for roots alone |
| Quadratic formula | Any quadratic equation, always works | None — universally applicable |
| Graphing | Visual/approximate solutions, checking work | Not precise for irrational roots |
Vertex, Axis of Symmetry & Graph
Every parabola has a vertex — its highest or lowest point — and a vertical axis of symmetry passing through it. The axis of symmetry is found with x = −b / (2a); substituting this x-value back into the original equation gives the vertex's y-coordinate. If a > 0, the vertex is the parabola's minimum point; if a < 0, it's the maximum point. The calculator above computes both automatically alongside the roots.
Real-World Uses
| Field | How Quadratic Equations Are Used |
|---|---|
| Physics | Projectile motion — height over time under gravity |
| Engineering | Structural curves, parabolic arches and reflectors |
| Business & economics | Profit maximization, revenue and cost modeling |
| Architecture | Designing parabolic shapes for bridges and domes |
| Sports | Modeling the trajectory of a thrown or kicked ball |
Common Mistakes
- Forgetting the ± sign. Every quadratic (with a nonzero discriminant) has two roots — only calculating one loses half the answer.
- Sign errors with negative b. Remember the formula uses −b, so if b is already negative, −b becomes positive — a very common slip.
- Stopping at a negative discriminant. A negative discriminant doesn't mean "no solution" — it means the solutions are complex numbers, not that they don't exist.
- Dividing only part of the numerator by 2a. Both the −b term and the entire ±√(...) term must be divided by 2a, not just one part.
- Misidentifying a, b, or c when the equation isn't in standard form. Always rearrange to ax² + bx + c = 0 first.
Limitations of This Calculator
- Solves single-variable quadratics only. This tool handles ax² + bx + c = 0 in one variable, not systems of equations or higher-degree polynomials.
- Requires a ≠ 0. If a = 0, the equation is linear, not quadratic, and falls outside this tool's scope.
- Displays complex roots in standard a ± bi form. Further complex-number operations are outside this calculator's scope.
- Numeric precision. Irrational roots are shown as rounded decimals; for exact radical form, simplify by hand using the calculator's discriminant as a starting point.
Glossary
- Quadratic Equation
- An equation of the form ax² + bx + c = 0 where a ≠ 0.
- Root (Solution)
- A value of x that makes the equation true; graphically, where the parabola crosses or touches the x-axis.
- Discriminant
- The expression b² − 4ac; its sign determines whether roots are real and distinct, real and repeated, or complex.
- Complex Root
- A solution involving the imaginary unit i (√−1), occurring when the discriminant is negative; complex roots always come in conjugate pairs (p ± qi).
- Vertex
- The turning point of a parabola — its minimum (if a > 0) or maximum (if a < 0) point.
- Axis of Symmetry
- The vertical line x = −b/(2a) passing through the vertex, about which the parabola is mirror-symmetric.
- Standard Form
- Writing a quadratic equation as ax² + bx + c = 0, with all terms on one side set equal to zero.
Frequently Asked Questions
Q: What is the quadratic formula?
A: x = (−b ± √(b² − 4ac)) / (2a) — it solves any equation of the form ax² + bx + c = 0.
Q: What is the discriminant and what does it tell you?
A: b² − 4ac. Positive means two real roots, zero means one repeated real root, negative means two complex roots.
Q: What if the quadratic formula gives a negative number under the square root?
A: The equation has two complex conjugate roots in the form p ± qi rather than real number solutions.
Q: How do you know an equation is quadratic?
A: It can be written as ax² + bx + c = 0 with a ≠ 0 — the highest power of the variable is exactly 2.
Q: What is the vertex of a parabola and how is it calculated?
A: Its x-coordinate is −b/(2a); substitute that back into the equation to get the y-coordinate.
Q: Can every quadratic equation be solved by factoring?
A: No — factoring only works for rational roots; the quadratic formula solves every quadratic, including irrational and complex root cases.
Q: What is the difference between the quadratic formula and completing the square?
A: They're mathematically equivalent — completing the square reveals the vertex form directly, while the formula is derived from it and is faster for just finding the roots.
This tool is provided for general educational use. For graded coursework, verify results against your course's required method and notation.