Quadratic Formula Calculator with steps
Solve any quadratic equation ax² + bx + c = 0 instantly. Get real or complex roots, discriminant analysis, factored form and a live interactive parabola graph — every step shown in clean, readable math.
Enter your coefficients
Fill in a, b and c from your equation in the form ax² + bx + c = 0.
Solution & Step-by-Step Working
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Built for accuracy, clarity and speed
Every feature is designed to help you actually understand the solution, not just see an answer.
Full Step-by-Step Solutions
See the discriminant, formula substitution and simplification laid out clearly with LaTeX-rendered math.
Real & Complex Roots
Negative discriminants automatically resolve to complex conjugate roots in a + bi form.
Interactive Parabola Graph
Visualise your equation's curve, vertex, axis of symmetry and root positions instantly.
Factoring Method
When roots are rational, get the fully factored form (x − p)(x − q) = 0 alongside the formula method.
Completing the Square
Follow the alternate algebraic technique with every transformation explained in order.
Copy, Download & Share
Export your worked solution as a text file, copy it to your clipboard, or share the tool directly.
Real-Time Validation
Instant feedback catches invalid entries (like a = 0) before you even hit calculate.
Dark & Light Themes
A polished interface that adapts to your preference and is easy on the eyes day or night.
Fully Responsive
Works flawlessly on mobile, tablet and desktop with no horizontal scrolling, ever.
How the quadratic formula calculator works
Four straightforward steps take you from raw coefficients to a fully explained solution.
Enter coefficients a, b, c
Type the numbers from your equation ax² + bx + c = 0. Decimals and negatives are supported.
Choose a solving method
Pick the standard formula, factoring, or completing the square — or compare all three.
Calculate instantly
The calculator computes the discriminant and both roots in real time, right in your browser.
Review, copy or share
Scroll through the full working, then copy, download, or share your result with one click.
Quadratic Equation Solver: A Complete Guide to the Quadratic Formula
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are real numbers and a is never zero. If a were zero, the x² term would disappear and the equation would collapse into a simple linear equation instead. This single family of equations shows up everywhere: in physics problems about falling objects, in business models that describe profit against price, in architecture where arches and cables trace parabolic curves, and in everyday classroom homework. Understanding how to solve them reliably is one of the most useful skills in algebra, and this quadratic formula calculator was built to make that process transparent rather than a black box.
What is the quadratic formula?
The quadratic formula is a general-purpose tool that solves any quadratic equation, no matter how messy the numbers look. Written out, it reads x equals negative b, plus or minus the square root of b squared minus four a c, all divided by two a. The expression underneath the square root, b² − 4ac, is called the discriminant, and it does a lot of the heavy lifting: its sign alone tells you whether the equation has two different real answers, one repeated answer, or a pair of complex answers. Because the formula works for every valid combination of a, b and c, it is often the fastest fallback when factoring looks difficult or impossible by inspection.
Solving a quadratic equation with steps
When you use a quadratic equation solver with steps, you are really watching three things happen in sequence. First, the calculator identifies a, b and c from your input and substitutes them directly into the formula. Second, it evaluates the discriminant, b² − 4ac, and simplifies the square root as far as possible, whether that produces a whole number, an irrational decimal, or an imaginary result. Third, it splits the plus-or-minus symbol into two separate calculations to produce the two roots, x₁ and x₂, and simplifies each fraction to its lowest terms. Seeing every one of these steps, rather than just a final answer, is what actually builds algebra fluency and helps you spot arithmetic mistakes in your own homework.
Solving a quadratic equation by factoring
A quadratic equation solver with steps by factoring takes a different route that can be quicker when the coefficients are small, whole numbers. Factoring rewrites ax² + bx + c as a product of two binomials, such as (x − p)(x − q), where p and q multiply to give a·c and add to give b. Once the expression is factored, the zero-product property says that if two factors multiply to zero, at least one of them must be zero, so you set each bracket equal to zero and solve for x separately. Factoring is elegant and fast, but it only works cleanly when the roots are rational numbers; for anything messier, the quadratic formula or completing the square remains the reliable choice, which is exactly why this tool offers both methods side by side.
Examples of the quadratic formula in practice
Consider x² − 5x + 6 = 0. Here a = 1, b = −5 and c = 6, so the discriminant is 25 − 24 = 1, a perfect square, giving two clean real roots: x = 2 and x = 3. It also factors neatly as (x − 2)(x − 3) = 0, confirming the same answer two different ways. Now take x² + 2x + 5 = 0: the discriminant is 4 − 20 = −16, a negative number, which means the roots are complex, specifically −1 + 2i and −1 − 2i. These worked examples of the quadratic formula illustrate why checking the discriminant first is such a useful habit before you commit to a full calculation.
| Discriminant (b²−4ac) | Nature of Roots | Example Equation |
|---|---|---|
| Positive | Two distinct real roots | x² − 5x + 6 = 0 |
| Zero | One repeated real root | x² − 4x + 4 = 0 |
| Negative | Two complex conjugate roots | x² + 2x + 5 = 0 |
How to use this quadratic formula calculator
Using the tool is simple: enter your three coefficients into the a, b and c fields above, choose whichever solving method you want to see explained, and press Calculate Roots. The page automatically validates your inputs in real time, flags common issues such as a equalling zero, and then scrolls you straight to a fully worked solution complete with a live parabola graph. From there you can copy the working to your clipboard, download it as a text file for your notes, or share the calculator with a classmate. Whether you are checking your own homework, studying for an exam, or just curious how a particular equation behaves, this quadratic equation solver calculator is designed to make every part of that process fast, transparent and genuinely useful.
Frequently asked questions
The five most common questions people ask about the quadratic formula.
The quadratic formula, x = (−b ± √(b²−4ac)) / 2a, solves any equation of the form ax²+bx+c=0. Plug in the coefficients a, b and c, compute the discriminant b²−4ac, then substitute it back into the formula to find both roots.
Factoring works by rewriting ax²+bx+c as a product of two binomials, such as (x−p)(x−q)=0, then setting each factor to zero. It is fastest when a=1 and the roots are rational, whole numbers.
The discriminant, b²−4ac, predicts the nature of the roots. A positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex conjugate roots.
Yes. When the discriminant is negative, the square root becomes imaginary, so the two roots take the form p ± qi, where i is the imaginary unit. This calculator handles that case automatically.
Quadratic equations model projectile motion, area optimisation, profit maximisation and satellite dish geometry. For instance, x²−5x+6=0 factors to (x−2)(x−3)=0, giving roots x=2 and x=3.
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