Solve ax² + bx + c = 0 with full working — roots, discriminant, vertex, axis of symmetry, factorisation and a graph.
The quadratic formula solves any equation of the form ax² + bx + c = 0. It states that x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is called the discriminant: if it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the roots are complex. Enter your coefficients below to see every step.
Enter the coefficients from ax² + bx + c = 0. Decimals and negatives are fine. If a is 0 the equation is linear, and the calculator will solve it as such.
Keyboard: Enter solves · Esc clears
The graph rescales automatically to show the vertex and any real roots clearly. It is a visual guide and may not use equal units on both axes.
A quadratic equation is any equation where the highest power of the unknown is 2. Written in standard form it looks like this:
The letters a, b and c are the coefficients. The condition a ≠ 0 matters: if a were 0 the x² term would vanish and the equation would be linear, with just one solution. Every quadratic graphs as a parabola, a symmetrical U-shaped curve, and its solutions are the points where that curve crosses the x-axis.
The ± sign is what produces two answers: one using the plus, one using the minus. The formula works for every quadratic, whether or not it factorises, which is why it is worth memorising.
Where it comes from. The formula is the result of completing the square on the general equation ax² + bx + c = 0. Rather than completing the square each time, mathematicians did it once with letters, and the formula is the answer.
The expression under the square root has its own name and its own job. The discriminant, written D or Δ, tells you what kind of roots to expect before you finish solving.
| Discriminant | Roots | Graph | Example |
|---|---|---|---|
| D > 0 | Two distinct real roots | Crosses the x-axis twice | x² − 5x + 6 → 3 and 2 |
| D = 0 | One repeated real root | Touches the x-axis once | x² + 2x + 1 → −1 |
| D < 0 | Two complex roots | Never touches the x-axis | x² + 2x + 5 → −1 ± 2i |
There is one more useful case: when D is a perfect square, the roots are rational and the quadratic factorises neatly with whole numbers. This calculator checks for that and shows the factorisation when it exists.
Enter a, b and c from your equation in standard form. If your equation is written differently — say 2x² = 5x − 3 — rearrange it so one side is zero first: 2x² − 5x + 3 = 0. The live display above the inputs shows the equation as you type, so you can check you have entered it correctly. Press Solve or the Enter key. You get the roots, the discriminant, the vertex, the axis of symmetry, the intercepts, a factorisation where one exists, a graph, and the full working.
| Feature | Factoring | Quadratic formula |
|---|---|---|
| Speed | Very fast when it works | Steady, always the same effort |
| Works when | Roots are rational | Always |
| Complex roots | Cannot find them | Handles them |
| Risk | Time wasted hunting for factors that do not exist | Arithmetic slips with signs |
| Best for | Neat exam questions | Anything real-world or messy |
A good strategy is to check the discriminant first. If it is a perfect square, factoring will work and is quicker. If not, go straight to the formula rather than searching for factors that cannot exist.
This third method rewrites the equation so the variable appears only once, which also converts it to vertex form. For x² + 6x + 5 = 0: halve the b coefficient to get 3, square it to get 9, then write x² + 6x + 9 − 9 + 5 = 0, giving (x + 3)² − 4 = 0. From there x + 3 = ±2, so x = −1 or −5.
Completing the square is how the quadratic formula itself is derived, and it is the fastest route to the vertex.
The vertex is the turning point — the minimum if a is positive and the parabola opens upward, or the maximum if a is negative and it opens downward. The axis of symmetry is the vertical line through the vertex, and the two roots always sit an equal distance either side of it.
| Feature | Standard form | Vertex form |
|---|---|---|
| Written | y = ax² + bx + c | y = a(x − h)² + k |
| Shows immediately | The y-intercept, c | The vertex, (h, k) |
| Best for | Applying the formula | Graphing and transformations |
| Example | y = x² − 4x + 3 | y = (x − 2)² − 1 |
| Feature | Real roots | Complex roots |
|---|---|---|
| Discriminant | D ≥ 0 | D < 0 |
| Form | Plain numbers | p ± qi, where i = √(−1) |
| On the graph | x-axis crossings | No crossings at all |
| Always occur | Individually | In conjugate pairs |
| Physical meaning | Real solutions exist | Often means "never happens" |
Complex roots are not a failure. In a projectile problem, complex roots mean the object never reaches that height. In electronics they describe oscillation, and in control systems they signal a system that rings rather than settling smoothly.
Projectile motion under gravity is quadratic in time, so roots give launch and landing.
h = −½gt² + v₀t + h₀Beam deflection, arch shapes and parabolic reflectors are all quadratic curves.
y = ax² + bx + cProfit often peaks at a vertex, so maximising revenue is a quadratic problem.
P = −ax² + bx − cSquared-error loss is quadratic, and gradient descent walks down that curve.
MSE = Σ(y−ŷ)²/nBézier curves and ray–sphere intersection both reduce to solving a quadratic.
at² + bt + c = 0Parabolic arches distribute load evenly, which is why they appear in bridges.
parabolic archA parabolic dish focuses every incoming signal to a single point.
y = x²/4fRisk–return optimisation and some pricing models minimise a quadratic function.
minimise variance1) x = 4 and 3 2) x = ±2 3) D = 1 4) x = −3 (repeated) 5) (2, −1)
1) x = 0.5 and −3 2) x = −2 ± 3i 3) x = 1 and −⅓ 4) k = ±6 (so that D = 0) 5) x = −2
What is the quadratic formula?
The quadratic formula solves any equation of the form ax squared plus bx plus c equals 0. It states that x equals minus b, plus or minus the square root of b squared minus 4ac, all divided by 2a.
What is a quadratic equation?
A quadratic equation is one where the highest power of the unknown is 2. In standard form it is written ax squared plus bx plus c equals 0, where a is not zero.
How do you solve a quadratic equation?
Rearrange it into standard form, identify a, b and c, then substitute them into the quadratic formula. You can also factorise or complete the square, but the formula always works.
What is the discriminant?
The discriminant is the part under the square root, b squared minus 4ac. It tells you how many roots there are and whether they are real or complex before you finish solving.
What does the discriminant tell you?
If it is positive there are two distinct real roots, if it is zero there is one repeated root, and if it is negative there are two complex roots. A discriminant that is a perfect square also means the equation factorises with whole numbers.
When does a quadratic have two roots?
When the discriminant is greater than zero. On a graph, the parabola crosses the x-axis at two separate points.
When does a quadratic have one root?
When the discriminant is exactly zero. The parabola touches the x-axis at a single point, which is also its vertex, and the root is described as repeated.
When does a quadratic have no real roots?
When the discriminant is negative. The parabola never reaches the x-axis, and the two solutions are complex numbers forming a conjugate pair.
What are complex roots?
Complex roots contain i, the square root of minus 1. They come in pairs such as minus 1 plus 2i and minus 1 minus 2i, and they occur whenever the discriminant is negative.
What is the vertex of a parabola?
The vertex is the turning point of the curve, found at x equals minus b divided by 2a. It is the lowest point when a is positive and the highest point when a is negative.
What is the axis of symmetry?
The axis of symmetry is the vertical line through the vertex, at x equals minus b over 2a. The two roots always sit an equal distance either side of it.
What is standard form?
Standard form is ax squared plus bx plus c equals 0. You need the equation in this shape before reading off the coefficients for the formula.
What is vertex form?
Vertex form is y equals a times x minus h, squared, plus k, where h and k give the vertex directly. It is more convenient than standard form for graphing.
How do you factor a quadratic?
Find two numbers that multiply to give a times c and add to give b, then split the middle term and group. Factoring only works neatly when the discriminant is a perfect square.
What is the difference between factoring and using the formula?
Factoring is quicker but only works when the roots are rational. The quadratic formula works for every quadratic, including those with irrational or complex roots.
What is completing the square?
Completing the square rewrites the equation so the variable appears only once, which also gives vertex form. It is the method from which the quadratic formula is derived.
Why can a not be zero?
If a were zero the x squared term would disappear and the equation would be linear rather than quadratic. The formula would also require dividing by 2a, which would be division by zero.
How do you find the x-intercepts?
The x-intercepts are the roots, the values of x where y equals 0. Solve the quadratic and each real root gives one crossing point on the graph.
How are quadratic equations used in physics?
Projectile motion under gravity is quadratic in time, so the roots give when an object leaves and returns to a given height. The vertex gives the maximum height reached.
How are quadratics used in engineering?
Engineers use them for beam deflection, arch design, parabolic reflectors and optimisation problems where a maximum or minimum sits at the vertex of a curve.
What are Vieta's formulas?
Vieta's formulas say the two roots add up to minus b over a and multiply to c over a. They give a quick way to check a solution without re-solving the equation.
Is this Quadratic Formula Calculator free?
Yes. It is free with no sign-up, works on any device, and runs entirely in your browser so your entries stay private.