Quadratic Equation Calculator
Enter the three coefficients of ax² + bx + c = 0 and this calculator returns the roots with full step-by-step working, the discriminant, the vertex, the axis of symmetry, and a graph of the parabola with the roots marked. Unlike most quadratic tools that print two roots and stop, this one offers four solution methods (quadratic formula, factoring with an honest fallback, completing the square, or all three side by side), decimal precision of 2, 3, 4 or 6 decimals plus an exact mode, and a built-in vertex-form reading of your equation.
Leave b or c blank to treat them as zero. The factoring method only applies when the roots are integers; the tool says so honestly when they are not.
Quick Answer
2x² + 5x − 3 = 0 has two distinct real roots: x₁ = 0.5 and x₂ = −3. The discriminant is D = 25 − 4(2)(−3) = 49, the vertex sits at (−1.25, −6.13), the axis of symmetry is x = −1.25, and the root sum and product (−2.5 and −1.5) match −b/a and c/a. Enter any a, b, c in the calculator above for the same full working, four solution methods, and a marked graph.
How to use this quadratic equation calculator
Start with the answer. Enter your three coefficients a, b and c (leave b or c blank and the tool treats them as zero, so x² − 4 = 0 and x² + 3x = 0 both work). Pick a solution method and a decimal precision, then press Solve equation. The tool checks first that a is a number and not zero; anything else returns the roots, the discriminant, the vertex, the axis of symmetry, the y-intercept, and the sum and product of the roots as built-in checks.
The method menu is where this tool earns its place. The quadratic formula always works; the default shows all methods side by side. Factoring is fastest when the roots are integers, but the tool says plainly when it does not apply instead of faking it: 2x² + 5x − 3 = 0 reports “Cannot factor with integer coefficients” because one root is 0.5, while x² − 5x + 6 = 0 factors cleanly into (x − 2)(x − 3). Completing the square rewrites the equation around the vertex, and “Show all methods” prints every applicable working side by side, which is the quickest way to compare them for homework.
Precision matters more than it looks. The default 2 decimals give x₁ = 0.50 and x₂ = −3.00; the exact mode strips rounding so 0.5 stays 0.5. If you are checking against a textbook that keeps fractions, exact mode is the one to use.
Worked example: 2x² + 5x − 3 = 0
The inputs above are pre-filled with this example, so pressing Solve equation reproduces it exactly.
Read off a = 2, b = 5, c = −3. The discriminant is D = 5² − 4(2)(−3) = 25 + 24 = 49. Because 49 is positive, there are two distinct real roots, and because it is a perfect square, factoring could in principle work, but the roots turn out not to be integers. Applying the quadratic formula: x = (−5 ± √49) / 4, so x₁ = 0.50 and x₂ = −3.00. The vertex is at x = −5/4 = −1.25, y = 2(1.5625) − 5(1.25) − 3 = −6.125, shown as (−1.25, −6.13) at the default precision. The axis of symmetry is the vertical line x = −1.25, the y-intercept is −3, and the sum of the roots (−2.5) and their product (−1.5) match −b/a and c/a. Completing the square gives (x + 1.25)² = 3.06, and the vertex-form reading under the result prints y = 2.00(x + 1.25)² − 6.13.
The quadratic formula
The quadratic formula solves any equation of the form ax² + bx + c = 0 with a ≠ 0: x = (−b ± √(b² − 4ac)) / (2a). The OpenStax Algebra and Trigonometry 2e textbook derives it by completing the square on the general equation, and its treatment of the discriminant is the standard reference. The expression under the square root, D = b² − 4ac, decides the story before you compute anything:
| Discriminant | Roots | What the graph does |
|---|---|---|
| D > 0 | Two distinct real roots | Crosses the x-axis twice |
| D = 0 | One repeated real root | Touches the x-axis at the vertex |
| D < 0 | Two complex conjugate roots | Never meets the x-axis |
The vertex of the parabola y = ax² + bx + c sits at x = −b/(2a), exactly halfway between the two real roots when they exist. That line is also the axis of symmetry. The sum of the roots is always −b/a and the product is always c/a, which is why the calculator prints both: they are a two-second sanity check on any answer.
Choosing a solution method
Four methods are offered because each reveals something different. The quadratic formula never fails, factoring teaches the structure of the roots when they are integers, and completing the square exposes the vertex directly in the form (x − h)² = k, where (h, k) is the vertex.
- Quadratic formula: the universal method; shows the substitution step by step.
- Factoring: fastest for integer roots such as x² − 5x + 6 = 0; reports honestly when the roots are not integers.
- Completing the square: rewrites the equation to read the vertex straight off; useful for graphing by hand.
- Show all methods: prints every applicable working so you can compare, which is what the example above uses.
If your equation factors into linear pieces and you want to explore that form further, the polynomial calculator goes deeper into polynomial structure. For equations that mix powers with logs rather than pure quadratics, the exponential and logarithmic equations calculator covers that ground.
Reading the graph
The graph plots y = ax² + bx + c with the vertex marked in red and each real root marked in green where the curve crosses the x-axis. When a is positive the parabola opens upward and the vertex is the minimum; when a is negative it opens downward and the vertex is the maximum. Complex roots mean the curve never touches the x-axis, which the graph makes obvious at a glance. The graph rescales itself around the vertex and roots, so a repeated root such as x² − 2x + 1 = 0 (vertex at (1, 0), touching the axis) and a wide-spaced pair both render cleanly.
What competing calculators miss
Most quadratic tools print two roots and stop: no vertex, no axis of symmetry, no method choice, no steps. The strongest rivals, including Symbolab with its method switcher and graph, and CalcWorld with its OpenStax-cited step-by-step working, each cover part of the territory. None of the free tools we checked combine all of it: four methods with an honest factoring fallback, precision from 2 decimals to exact mode, the discriminant read-out with root-nature labels, the vertex, axis of symmetry, y-intercept, the sum and product of roots as built-in answer checks, a vertex-form reading of the equation, and a rendered parabola with the vertex and roots marked. One rival offers a precision select; another offers method choice; this page is the only one that puts the whole set together, and the conversion work shows in the worked example above.
Frequently asked questions
Identify a, b and c in ax² + bx + c = 0, compute the discriminant b² − 4ac, then apply x = (−b ± √D) / (2a). The calculator does all of this and shows every step, for the quadratic formula, factoring, completing the square, or all three at once.
Whether the roots are real or complex before you find them. A positive discriminant means two distinct real roots, zero means one repeated real root (the parabola touches the x-axis at its vertex), and a negative discriminant means two complex conjugate roots, in which case the graph never crosses the x-axis.
Yes. When the discriminant is negative, the roots are a conjugate pair, for example x² + 2x + 5 = 0 gives x₁ = −1 + 2i and x₂ = −1 − 2i. The tool reports both parts with the selected precision, and the graph shows the parabola floating clear of the x-axis.
Choose the quadratic formula or completing the square instead. The factoring method here only claims success for integer roots, so for 2x² + 5x − 3 = 0 (roots 0.5 and −3) it reports that factoring with integer coefficients is not possible, while x² − 5x + 6 = 0 factors cleanly into (x − 2)(x − 3).
The turning point of the curve, at x = −b/(2a). It is the minimum when a is positive and the maximum when a is negative, and the vertical line through it is the axis of symmetry. The calculator also prints a vertex-form reading, such as y = 2.00(x + 1.25)² − 6.13, which shows the vertex directly.
Use the root relationships: the sum of the roots always equals −b/a and the product always equals c/a. The calculator prints both next to the roots, so for 2x² + 5x − 3 = 0 you can see at a glance that 0.5 + (−3) = −2.5 and 0.5 × (−3) = −1.5. If your hand-computed roots fail this check, one of them is wrong. For pure radical cases, the cube-root calculator handles the simpler one-step radical.
It skips decimal rounding entirely and prints the shortest representation: integers print without decimals and other values print up to 6 decimals with trailing zeros stripped, so 0.50 becomes 0.5 and the vertex of 2x² + 5x − 3 = 0 shows as (−1.25, −6.125) instead of (−1.25, −6.13). Use it when you need the tool’s raw values for further calculation.
Add This Calculator to Your Website
Embed this calculator on your own site with the code below. Pick a size to preview it live, then copy the code.
Powered by Calculator Zilo
References
- OpenStax, Rice University. 2.5 Quadratic Equations, Algebra and Trigonometry 2e. Accessed October 24, 2026.
- OpenStax, Rice University. 10.3 Solve Quadratic Equations Using the Quadratic Formula, Elementary Algebra 2e. Accessed October 24, 2026.
- Khan Academy. Quadratic formula explained (article). Accessed October 24, 2026.
- Khan Academy. Proof of the quadratic formula. Accessed October 24, 2026.
- Khan Academy. Number of solutions of quadratic equations (nature of roots). Accessed October 24, 2026.