Quadratic Formula Calculator
Enter the coefficients a, b, and c to solve ax² + bx + c = 0, with real or complex roots, the discriminant, the nature of the roots, and the parabola vertex.
Coefficients of ax² + bx + c = 0
Enter the three coefficients from your equation. Set a to zero to solve a linear equation. Everything runs in your browser. Nothing is uploaded.
Decimal places
Solution
Everything is calculated in your browser. Nothing is uploaded.
How to solve a quadratic equation online
Enter the coefficients
Type the values of a, b, and c from your equation ax² + bx + c = 0 into the three fields.
Read the roots
The two roots appear instantly, shown as real numbers or as complex conjugates when the discriminant is negative.
Check the discriminant and vertex
See the discriminant, the nature of the roots, and the parabola vertex update the moment a coefficient changes.
Copy the solution
Use the copy button to place a plain summary of the roots, discriminant, and vertex on your clipboard.
Why use this tool
Real and complex roots
When the discriminant is negative the two roots are shown as complex conjugates written with the letter i, not reported as an error.
Discriminant and root nature
The discriminant b² − 4ac is computed and labelled, telling you at a glance whether the equation has two, one, or no real roots.
Vertex, vertex form, and factored form
The turning point of the parabola and its axis of symmetry come from the same coefficients, along with the vertex form and, for real roots, the factored form.
Handles a = 0 and repeated roots
Set a to zero and the equation is solved as a linear one, while a zero discriminant is reported as a single repeated root.
Adjustable precision
Switch between 2, 4, 6, and 8 decimal places to keep irrational roots as short or as exact as you need.
Runs entirely in your browser
Every calculation happens on your device. Nothing you type is uploaded.
About this tool
A quadratic equation has the form ax² + bx + c = 0, and the quadratic formula gives its solutions from the three coefficients alone. This calculator applies that formula the moment you enter a, b, and c. It returns both roots, the discriminant that decides how many real roots exist, the nature of those roots, and the vertex of the matching parabola, all updated instantly with no button to press.
It handles the cases a plain formula skips over. When the discriminant b² − 4ac is negative the roots are complex, so they are shown as a conjugate pair written with the letter i rather than as an error. A zero discriminant is reported as one repeated root, and if you set a to zero the equation becomes linear and is solved as bx + c = 0. The square root at the heart of the formula is often irrational, so you can choose how many decimal places to show, and for that root on its own the square root calculator goes further.
Alongside the roots you get the axis of symmetry, the vertex form, and, when the roots are real, the factored form of the expression, which together describe the whole parabola. For raising the coefficients to powers try the exponent calculator, and for longer mixed calculations the scientific calculator covers trigonometry, logarithms, and more.
Frequently asked questions
- How does the quadratic formula calculator work?
- Enter the three coefficients a, b, and c and the calculator substitutes them into the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. Both roots, the discriminant, the nature of the roots, and the vertex appear instantly with no button to press.
- What happens when the discriminant is negative?
- A negative discriminant means the equation has no real roots, so the two solutions are complex conjugates. They are shown in the form p ± qi using the letter i for the imaginary unit, instead of being reported as an error.
- Can it solve an equation where a is zero?
- Yes. If a is zero the equation is not quadratic but linear, so it is solved as bx + c = 0 to give the single root −c ÷ b. The discriminant and vertex do not apply and are marked as such.
- What is the discriminant and why does it matter?
- The discriminant is b² − 4ac. When it is positive there are two distinct real roots, when it is zero there is one repeated real root, and when it is negative the roots are complex. The calculator labels which case applies.
- Why do the roots have so many decimal places?
- The formula contains a square root, which is usually irrational, so the roots rarely end in a neat number. Values are rounded to the number of decimal places you choose, from 2 up to 8.
- Is my data uploaded anywhere?
- No. Everything runs in your browser; nothing is sent to a server.
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