Solve any quadratic equation, with the working shown
This quadratic equation solver finds the roots of any equation of the form ax² + bx + c = 0. Type the three coefficients and you instantly get both roots — as exact values such as 1 ± √2 or −1 ± 2i and as decimals — together with the discriminant, the vertex, the axis of symmetry, the y-intercept and the factored form whenever the roots are rational.
Every answer comes with a step-by-step solution using the quadratic formula, written the way teachers expect to see it in homework and exams, plus a graph of the parabola with its roots and vertex marked. Coefficients can be whole numbers, decimals or fractions, and the arithmetic is done with exact fractions, so you will not see rounding errors like 0.49999999 in place of ½.
How to use the Quadratic Equation Solver
- Rearrange your equation so one side is zero, for example 2x² + 3x = 2 becomes 2x² + 3x − 2 = 0.
- Enter a (the number in front of x²), b (the number in front of x) and c (the constant). Use a minus sign for negative values, and 0 for any missing term.
- Read the roots in the result panel — exact form in the caption, decimals in large type.
- Check the details table for the discriminant, vertex, axis of symmetry, factored form and vertex form.
- Follow the numbered step-by-step solution, or press “Copy solution” to paste it into your notes.
Quadratic Equation Solver features
Exact surd and complex roots
Square roots are simplified (√12 = 2√3) and complex roots are written with i, not just rounded decimals.
Factored form
When the roots are rational, the equation is factorised for you, for example 2x² + 3x − 2 = (2x − 1)(x + 2).
Vertex and graph
See the vertex, axis of symmetry, y-intercept and a crisp plot of the parabola that follows the light or dark theme.
Fractions and decimals
Enter coefficients such as 0.5 or 3/4. The solver works with exact fractions internally.
Numerically stable decimals
Decimal roots use the cancellation-free form of the formula, so even equations like x² − 10⁶x + 1 = 0 give accurate small roots.
Helpful error messages
If a = 0 the solver explains that the equation is linear and still gives you its solution.
The quadratic formula and the discriminant
Every quadratic equation ax² + bx + c = 0 (with a ≠ 0) is solved by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, Δ = b² − 4ac, is called the discriminant, and it tells you what kind of roots to expect before you finish the calculation.
Geometrically, the roots are the points where the parabola y = ax² + bx + c crosses the x-axis. A positive discriminant means the curve crosses twice, zero means it just touches the axis at the vertex, and a negative discriminant means it never reaches the axis at all.
| Discriminant | Roots | Graph | Example |
|---|---|---|---|
| Δ > 0, perfect square | Two different rational roots | Crosses the x-axis twice | x² − 5x + 6 = 0 → 2, 3 |
| Δ > 0, not a perfect square | Two different irrational roots | Crosses the x-axis twice | x² − 2x − 1 = 0 → 1 ± √2 |
| Δ = 0 | One repeated real root | Touches the x-axis at the vertex | 4x² − 4x + 1 = 0 → ½ |
| Δ < 0 | Two complex conjugate roots | Never meets the x-axis | x² + 2x + 5 = 0 → −1 ± 2i |
Worked example: 2x² + 3x − 2 = 0
Here a = 2, b = 3 and c = −2. The discriminant is Δ = 3² − 4 × 2 × (−2) = 9 + 16 = 25, a perfect square, so the roots are rational. Using the formula, x = (−3 ± √25) / (2 × 2) = (−3 ± 5) / 4, which gives x₁ = 2/4 = ½ and x₂ = −8/4 = −2.
Because the roots are ½ and −2, the factored form is (2x − 1)(x + 2). The vertex sits halfway between the roots at x = −b/2a = −¾, and substituting back gives y = −25/8, so the vertex is (−0.75, −3.125). You can check the roots with Vieta’s formulas: their sum is −b/a = −1.5 and their product is c/a = −1.
Common mistakes when solving quadratics
If you need to simplify a radical on its own, the square root calculator shows the steps, and the fraction calculator helps when the coefficients are fractions. For right-triangle problems that lead to quadratics, try the Pythagorean theorem calculator.
- Forgetting to set the equation to zero. x² + 3x = 10 must become x² + 3x − 10 = 0 before you read off c = −10.
- Dropping the sign of b. The formula starts with −b, so when b is negative, −b is positive. Writing b in brackets, (−b), avoids this slip.
- Squaring a negative b incorrectly. (−4)² = 16, not −16. The discriminant’s b² is never negative.
- Dividing only part of the numerator by 2a. The whole expression −b ± √Δ is divided by 2a, not just the square root.
- Stopping at “no solution” when Δ < 0. There are no real roots, but there are two complex roots — useful in physics, engineering and higher maths.
Quadratic Equation Solver FAQ
How do I use the quadratic equation solver?
Rearrange your equation to the form ax² + bx + c = 0, then type a, b and c. The roots, discriminant, vertex, factored form, graph and full working appear immediately.
What does the discriminant tell me?
The discriminant Δ = b² − 4ac decides the type of roots: positive means two real roots, zero means one repeated root, and negative means two complex conjugate roots with no real solution.
Can the solver handle complex (imaginary) roots?
Yes. When the discriminant is negative, the roots are shown in the form p ± qi, both exactly (for example −1 ± 2i or (−1 ± i√3)/2) and as decimals.
Why does the solver say a cannot be 0?
If a = 0 the x² term disappears and the equation becomes linear (bx + c = 0), which has at most one solution. The solver shows that solution, x = −c/b, instead.
How do I find the vertex of a parabola?
The vertex is at x = −b / 2a. Substitute that x back into the equation to find y. The solver shows the vertex both exactly and as decimals, together with the vertex form a(x − h)² + k.
When can a quadratic be factored?
A quadratic with rational coefficients factors neatly over the rational numbers exactly when its discriminant is a perfect square. The solver then shows the factored form, such as (2x − 1)(x + 2).