Quadratic Equation Calculator
Solve any quadratic equation instantly. Enter the coefficients a, b, and c to find real or complex roots, the discriminant, vertex, axis of symmetry, and step-by-step working. Free, private, and runs entirely in your browser.
Enter Coefficients
Solution
Enter the values of a, b, and c, then click Solve Equation to see the roots, discriminant, vertex, and step-by-step working.
What does this quadratic equation calculator do?
This free online quadratic equation calculator solves any equation of the form ax² + bx + c = 0. Just enter the three coefficients — a, b, and c — and the calculator instantly finds the roots, whether they are real or complex. It also computes the discriminant, the vertex of the parabola, the axis of symmetry, and the sum and product of the roots.
Unlike a basic equation solver, this tool shows the full working. You can see how the coefficients are substituted into the quadratic formula, how the discriminant is calculated, and how the roots are simplified. This makes it useful not just for getting the answer, but for understanding the process.
Everything runs locally in your browser. Your coefficients never leave your device, there's no signup, and there are no usage limits. It's built for students, teachers, engineers, and anyone who needs to solve a quadratic equation quickly and accurately.
The quadratic formula explained
The quadratic formula is one of the most important tools in algebra. It gives the solutions to any quadratic equation directly from the coefficients:
x = −b ± √(b² − 4ac)2a
You can use it whenever a quadratic equation cannot be solved easily by factoring. It works for every quadratic equation — including ones with irrational or complex roots.
The discriminant decides everything
The expression under the square root, b² − 4ac, is called the discriminant. Its value tells you what kind of roots to expect before you even finish the calculation:
| Discriminant (D) | Nature of roots | Example |
|---|---|---|
D > 0 |
Two distinct real roots | x² − 5x + 6 = 0 → x = 2, 3 |
D = 0 |
One real root (repeated) | x² − 4x + 4 = 0 → x = 2 |
D < 0 |
Two complex conjugate roots | x² + x + 1 = 0 → x = −½ ± i√3⁄2 |
Worked example: x² − 5x + 6 = 0
Here a = 1, b = −5, c = 6.
- Compute the discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1
- Since D > 0, there are two distinct real roots.
- Substitute into the formula: x = (5 ± √1) ÷ 2
- Root 1: x = (5 + 1) ÷ 2 = 3
- Root 2: x = (5 − 1) ÷ 2 = 2
Factoring confirms this: x² − 5x + 6 = (x − 2)(x − 3), so the roots are indeed 2 and 3.
Why use this quadratic equation calculator?
Instant solution
Enter the coefficients and get the roots immediately. No manual substitution, no arithmetic errors.
Real and complex roots
Handles all three cases: two real roots, one repeated root, and two complex conjugate roots with the imaginary unit i.
Step-by-step working
See how the discriminant is computed and how the quadratic formula is applied, with each step shown clearly.
Vertex and axis of symmetry
Also computes the vertex (x, y) of the parabola and the equation of its axis of symmetry.
100% private
Everything runs in your browser. Your coefficients never leave your device — no server, no logs, no tracking.
Works offline
Once the page loads, you can solve equations without an internet connection. Bookmark it for homework sessions.
Who is this quadratic equation calculator for?
Students
- Solve homework problems in algebra and precalculus
- Check your answers after solving by factoring or completing the square
- Understand the link between the discriminant and the nature of the roots
- Study for exams by comparing your working with the step-by-step solution
Teachers and tutors
- Demonstrate the quadratic formula and discriminant in class
- Quickly generate example equations with specific root types
- Verify answer keys without doing every calculation manually
Engineers and scientists
- Solve equations that appear in physics problems involving motion and energy
- Find the intersection points of parabolas and lines in design work
- Compute vertex values for optimisation problems
Everyday users
- Work out projectile motion problems for DIY or sports calculations
- Solve area and dimension problems that reduce to quadratics
- Check quadratic relationships in budgeting and cost analysis
Real-world applications of quadratic equations
Quadratic equations are not just abstract algebra — they show up in many real situations:
- Projectile motion: The height of a ball, rocket, or thrown object over time follows a quadratic equation. Solving it tells you when the object hits the ground or reaches its peak.
- Area problems: Finding the dimensions of a rectangle with a given perimeter and area often leads to a quadratic equation.
- Profit and revenue: In business, revenue and profit functions are often quadratic. Their vertex gives the price or quantity that maximises profit.
- Physics: Equations for kinetic energy, gravitational potential, and motion under constant acceleration are quadratic in form.
- Engineering: Parabolic shapes appear in suspension bridge cables, satellite dishes, car headlights, and architectural arches.
- Accident reconstruction: Traffic investigators use quadratic equations to estimate vehicle speed from skid-mark length and road friction.
Tips for solving quadratic equations
- Put the equation in standard form first. Rearrange everything so the equation reads ax² + bx + c = 0 before identifying a, b, and c. A missing term means its coefficient is 0.
- Check for easy factoring. If the numbers are small integers, try factoring before reaching for the formula. x² − 5x + 6 factors as (x − 2)(x − 3) in seconds.
- Compute the discriminant early. Calculating D = b² − 4ac first tells you what kind of roots to expect, so you know whether to write real numbers or include i.
- Watch the signs. The most common mistakes come from mishandling negative numbers, especially when b is negative and you substitute −b.
- Simplify the root. If √D is not a perfect square, simplify it by pulling out perfect-square factors. √12 becomes 2√3, for example.
- Verify with Vieta's formulas. The sum of the roots should equal −b ÷ a and the product should equal c ÷ a. This is a quick sanity check on your answer.
- Remember the vertex tells you the extreme value. If a > 0, the vertex is the minimum of the parabola; if a < 0, it's the maximum. This is essential for optimisation problems.