Quadratic Equation Solver
Paste an equation or enter a, b, c. Solve to see roots and steps.
Steps for this equation
Supported notation
Equation mode accepts x, x² or x^2, + and −, decimal coefficients, and optional multiplication signs: 2*x^2 − 8 = 0. An expression without = is set equal to zero. Brackets, fractions, functions, and scientific notation are not supported in equation mode; use decimal coefficients in a, b, c mode instead. This is a numeric solver, not a general algebra parser.
How to use the quadratic equation calculator
Enter an equation such as x² − 3x + 2 = 0, or choose a, b, c and enter 1, −3, and 2. Press Solve equation. The first answer identifies the type of solution and displays the roots; the working below uses your current coefficients. Editing or resetting clears the previous answer until you solve again.
The quadratic formula and standard form
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. Its roots are given by x = (−b ± √(b² − 4ac)) / (2a). The ± sign represents two choices. The discriminant, b² − 4ac, tells you whether those choices give two distinct real roots, one repeated real root, or a pair of complex roots.
Move every term to the same side before identifying the coefficients. For example, 2x² = 8 has a = 2, b = 0, and c = −8. A missing x term means b is zero; a missing constant means c is zero. Do not discard a minus sign when entering a coefficient.
Three examples: real, repeated, and complex roots
| Equation | Discriminant | Solutions |
|---|---|---|
| x² − 3x + 2 = 0 | 1 | x = 1 or 2 |
| x² − 2x + 1 = 0 | 0 | x = 1, repeated |
| x² + 1 = 0 | −4 | x = i or −i |
For the first example, x = (3 ± √1) / 2, giving 1 and 2. In the repeated-root example the parabola touches the x-axis at its vertex. In the complex example, i² = −1 and there are no real x-intercepts.
Reading the graph and vertex
The graph shows y = ax² + bx + c over a finite, automatically chosen window. Orange points are real roots, where y = 0. The blue V identifies the vertex. For a quadratic, its x-coordinate is −b/(2a), which also gives the vertical axis of symmetry. The vertex is a minimum when a is positive and a maximum when a is negative. The axes can use different scales, so the picture is a guide rather than a measurement drawing.
Numerical method and limits
The calculator scales coefficients to reduce overflow and uses a cancellation-safe evaluation for distinct real roots. These protections improve numerical behavior but do not make floating-point arithmetic exact. Decimal results are rounded for display. For safe integer coefficients and |b² − 4ac| ≤ 1,000,000,000, a separate exact calculation reduces fractions and simplifies radicals. For instance, 2x² − 4 = 0 gives ±√2, while x² − 8x + 5 = 0 gives 4 ± √11. Larger discriminants and noninteger coefficients retain the substituted formula and approximate roots; they are not silently labeled simplified. If the original discriminant is too large to display but a scaled solution is valid, the answer explains the scaling. A graph may be unavailable even when the roots are valid.
Frequently asked questions
How do I solve a quadratic equation?
Put the equation in ax² + bx + c = 0 form, with a not equal to zero. Substitute the coefficients into x = (−b ± √(b² − 4ac)) / (2a). This calculator accepts an equation or the three coefficients, then shows the discriminant, substituted formula, numerical roots, and vertex.
What does the discriminant tell me?
For real coefficients and a ≠ 0, the discriminant Δ = b² − 4ac determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots. Negative does not mean there are no complex solutions.
Can I paste a complete equation?
Yes. Enter terms in x using x² or x^2, signs, decimals, and an optional equals sign. Both sides are moved into standard form. For example, 2x² = 8 becomes 2x² − 8 = 0. Brackets, fractions, functions, and scientific notation are not accepted in equation mode; the coefficient mode accepts signed decimals and scientific notation.
What if a is zero?
The equation is not quadratic. If b is nonzero, it is linear and its solution is x = −c/b. If a and b are both zero, c = 0 is an identity and every real number works; a nonzero c gives no solution.
Are the roots exact?
For safe integer coefficients with |b² − 4ac| no larger than 1,000,000,000, the tool shows simplified exact roots, including fractions, radicals and complex pairs. For example, x² − 8x + 5 = 0 gives 4 ± √11. Outside that range it retains the substituted formula and numerical roots. Numerical roots and vertex coordinates are floating-point approximations, displayed to at most 12 significant digits; this is not exact decimal arithmetic.
Why does a quadratic with complex roots have no x-intercepts?
An x-intercept has a real x-coordinate and y = 0. A quadratic with two non-real complex roots does not cross or touch the real x-axis. Its real-coordinate graph still has a vertex and opens upward when a > 0 or downward when a < 0.