About Quadratic Formula Calculator
This quadratic formula calculator is free, needs no sign-up and runs in your browser, so the numbers you type are never uploaded. It solves ax² + bx + c = 0 exactly: whole-number and fraction roots stay exact, irrational roots are shown as simplified radicals such as -1 + √2, and negative discriminants give the complex pair instead of an error.
Beside the roots you get the discriminant with a plain-English reading, the vertex and axis of symmetry, the factored form, the vertex form, and a plot of the parabola with the roots and vertex marked. Pick the view you need: Roots, Discriminant, Vertex or Completing the square. The same inputs feed all four.
Coefficients may be integers, decimals or fractions like 1/2. You can also type a whole equation such as 3x^2 - 5x + 2 = 0 and the calculator reads the coefficients for you. Every result comes with the working written in KaTeX, so you can copy the method into homework or check your own answer.
How to use Quadratic Formula Calculator
- Choose what to show: Roots, Discriminant, Vertex or Completing the square.
- Enter a, b and c in their boxes, or switch to Equation and type something like 2x^2 - 7x + 3 = 0.
- Read the headline answer. It updates as you type, with no calculate button.
- Open the steps underneath to follow the substitution into the quadratic formula, or the completing-the-square route.
- Use the table for the factored form, vertex form, axis of symmetry and y-intercept, and the graph to see where the parabola crosses the x axis.
Features
- Exact roots: rational, simplified radical or complex, never only a rounded decimal.
- Discriminant with a reading: two real roots, one repeated root or a complex pair.
- Vertex, axis of symmetry, y-intercept and whether the parabola opens up or down.
- Factored form over the rationals when it exists, with the complex factorisation otherwise.
- Completing the square written out, ending in vertex form.
- Quadratic formula steps in KaTeX, with a copy button for the LaTeX.
- Interactive plot with roots and vertex marked, zoom, pan and PNG export.
- Accepts integers, decimals, fractions and a full equation.
The quadratic formula and the discriminant
For ax² + bx + c = 0 with a not equal to 0, the roots are given by the quadratic formula. The expression under the root, b² - 4ac, is the discriminant. Its sign tells you the kind of roots before you finish the arithmetic.
Positive discriminant: two different real roots. Zero: one repeated real root, and the parabola just touches the x axis. Negative: no real roots, and the roots are a complex conjugate pair.
x = (-b ± √(b² - 4ac)) / (2a)
Δ = b² - 4ac
vertex: x = -b / (2a), y = c - b² / (4a)Worked examples
2x² - 7x + 3 = 0 gives x = 3 and x = 1/2
The discriminant is 25, a perfect square, so both roots are rational. The vertex is at x = 7/4.
x² + 2x - 1 = 0 gives x = -1 + √2 and x = -1 - √2
The discriminant is 8, not a perfect square, so √8 simplifies to 2√2 and the roots stay as radicals: about 0.4142 and -2.4142.
x² + 2x + 5 = 0 gives x = -1 + 2i and x = -1 - 2i
The discriminant is -16, so the parabola never reaches the x axis. The roots are the complex pair -1 ± 2i.
x² - 6x + 9 = 0 gives x = 3 (repeated)
The discriminant is 0, so the equation is a perfect square, (x - 3)² = 0.
What the discriminant tells you
| Discriminant | Roots | Graph |
|---|---|---|
| Positive | Two different real roots | Crosses the x axis twice |
| Positive perfect square | Two rational roots, so it factors over the integers | Crosses at two rational points |
| Zero | One repeated real root | Touches the x axis at the vertex |
| Negative | Two complex roots, a conjugate pair | Never meets the x axis |
Good to know
Limits of this calculator
It solves quadratics in one variable with real coefficients. If a is 0 the equation is linear and the calculator tells you so rather than guessing. Equations with brackets must be expanded first.
Exactness depends on the coefficients being rational. Decimals are read as exact decimal fractions, so 0.1 is treated as 1/10 and not as a floating-point number.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The coefficients you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- Check an answer by substituting a root back into the equation: the left side should come out as 0.
- If the roots look awkward, look at the discriminant first. A perfect square means rational roots and a clean factorisation.
- The sum of the roots is -b/a and their product is c/a, which is a quick check on any pair of roots.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Coefficients are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Coefficients are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is the quadratic formula?
For ax² + bx + c = 0 the solutions are x = (-b ± √(b² - 4ac)) / (2a). The calculator substitutes your a, b and c into it and simplifies each step.
What does a negative discriminant mean?
It means there are no real roots, because you would need the square root of a negative number. The calculator gives the two complex roots, which are conjugates of each other.
Can it give exact answers instead of decimals?
Yes. Rational roots appear as integers or fractions, and irrational roots as simplified radicals like -1 + √2. A decimal approximation is given only on the graph and in the notes.
How do I find the vertex of a parabola?
The x coordinate is -b/(2a) and the y coordinate is the value of the quadratic there. Choose the Vertex view to see both, plus the axis of symmetry and vertex form.
What happens if a is zero?
Then the equation is linear, not quadratic, and has at most one root. The calculator shows a message instead of dividing by zero.
Can I enter fractions or decimals?
Yes. Type 1/2 or 0.75 in any box. They are converted to exact fractions, so the roots stay exact.
What is completing the square?
It rewrites the quadratic as a perfect square plus a constant, a(x - h)² + k, which shows the vertex directly and leads to the same roots as the formula. The Completing the square view writes out each step.
Popular Quadratic Formula Calculator pages
- Discriminant CalculatorEnter a, b and c to get the discriminant and what it says about the roots.
- Vertex Form CalculatorEnter a, b and c to get the vertex, the axis of symmetry and the vertex form.
- Completing the Square CalculatorEnter a, b and c to see the equation rewritten as a perfect square, step by step.
- Quadratic Equation With Complex RootsSolve a quadratic with no real roots and get the complex pair, with the working.