Quadratic Formula Calculator
Solve quadratic equations with discriminant and roots
Quadratic Formula Calculator tool
1x² + -5x + 6 = 0
Results
Discriminant
—
Type
Two real roots
Roots:
x₁ =
x₂ =
Formula:
x = (-b ± √(b² - 4ac)) / 2a
x = (5 ± √(25 - 24)) / 2
x = (5 ± √null) / 2
x₁ =
x₂ =
Step-by-step:
- 1.Identify coefficients: a=1, b=-5, c=6
- 2.Calculate discriminant: b² - 4ac
- 3.Discriminant = (-5)² - 4(1)(6) = 0
- 4.Since discriminant < 0: complex conjugate roots
- 5.Apply quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
- 6.x = (5 ± √0) / 2
- 7.x₁ =
- 8.x₂ =
Quadratic Formula Calculator: key facts
- What it does
- Solve quadratic equations with discriminant and roots
- Category
- Math Calculators
- Cost
- Free, with no account, sign-up, or install.
- Your data
- Runs entirely in your browser — the files and text you enter are never uploaded to a server.
- Last reviewed
- . Report an incorrect result.
What this tool is for
The quadratic formula solves any equation of the form ax² + bx + c = 0, and this calculator applies it in full — including the cases where the answers are repeated or complex. Enter the three coefficients and it returns the roots, the discriminant, the vertex, and the axis of symmetry.
It is a constant companion in algebra and precalculus, where solving quadratics and analyzing parabolas is bread-and-butter work. By computing the discriminant first, it correctly identifies whether you get two real roots, one repeated root, or a complex conjugate pair — and it shows the complex answers properly rather than just failing when the discriminant is negative.
How to use the Quadratic Formula Calculator
- Enter coefficient a (it cannot be zero, or the equation is not quadratic).
- Enter coefficients b and c.
- Read the roots, with complex roots shown in proper a + bi form when the discriminant is negative.
- Check the discriminant, vertex, and axis of symmetry, along with the worked formula.
The discriminant decides everything
The tool applies the quadratic formula, and the quantity under the square root — the discriminant, b² − 4ac — determines the character of the solution before any root is taken. Positive gives two distinct real roots, zero gives one repeated root where the parabola just touches the axis, and negative gives a conjugate pair of complex roots with no real solution.
Because that single value settles the shape of the answer, the tool reports it explicitly and classifies the result rather than simply printing numbers. For a negative discriminant it separates the real and imaginary parts instead of failing.
Alongside the roots it computes the vertex and the axis of symmetry. The axis sits at −b over 2a, exactly midway between the roots when they are real, and the vertex is the parabola's turning point — the minimum when a is positive, the maximum when negative.
- The defaults — x² − 5x + 6 = 0 — have a discriminant of 1, giving roots at x = 3 and x = 2.
- The vertex sits at (2.5, −0.25) with the axis of symmetry at x = 2.5, midway between those roots.
- Changing c to 10 makes the discriminant −15, so the roots become complex and the parabola never crosses the axis.
What the Quadratic Formula Calculator gets right
- It handles all three discriminant cases — two real roots, one repeated root, and complex conjugate roots — instead of breaking on a negative discriminant.
- It computes the vertex and axis of symmetry too, so you can analyze the parabola, not just solve for x.
- It shows the discriminant and the full formula substitution, making the method clear.
- It runs locally with no account.
Factoring, completing the square, and numerical care
The formula always works, which is its virtue, but it is not always the quickest route. Where the roots are simple integers, factoring by inspection is faster — x² − 5x + 6 factors to (x − 2)(x − 3) almost on sight. Completing the square is the method the formula is derived from and is worth knowing because it produces the vertex form directly, which is more useful than the roots for graphing or optimisation.
A coefficient a of zero makes the equation linear rather than quadratic, and the formula breaks down by dividing by zero. That is a definitional boundary rather than a limitation.
One numerical caution: when b² is much larger than 4ac, one of the two subtractions cancels most of the significant digits and the corresponding root loses precision badly. Numerical libraries avoid this by computing the well-conditioned root first and obtaining the other from the fact that the product of the roots equals c over a. It rarely matters for textbook coefficients and matters a great deal in engineering computation.
Frequently Asked Questions
What is the discriminant and why does it matter?
The discriminant is b² − 4ac, the part under the square root in the quadratic formula. Its sign tells you the nature of the roots: positive gives two real roots, zero gives one repeated root, and negative gives a pair of complex roots.
What happens when the discriminant is negative?
The equation has no real solutions, but it has two complex conjugate roots. This calculator computes them and displays them in the form a + bi rather than simply reporting "no solution".
What are the vertex and axis of symmetry?
The vertex is the turning point of the parabola the equation describes, and the axis of symmetry is the vertical line through it, at x = −b/(2a). The calculator reports both so you can sketch or analyze the curve.
Why must coefficient a be non-zero?
If a is zero, the x² term disappears and the equation is linear, not quadratic — and the formula would divide by zero. The calculator requires a non-zero a; for a linear equation, solve bx + c = 0 directly instead.
