Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 for its real or complex roots, with the discriminant, vertex and axis of symmetry shown alongside.
Roots of ax² + bx + c = 0
x = 3 or x = 2
- Nature of roots
- Two distinct real roots
- Discriminant (b² − 4ac)
- 1
- Vertex x
- 2.5
- Vertex y
- -0.25
- Axis of symmetry
- x = 2.5
- Sum of roots (−b/a)
- 5
- Product of roots (c/a)
- 6
- Parabola opens
- Upward (minimum at vertex)
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any ax² + bx + c = 0 with a ≠ 0. The discriminant b² − 4ac controls how many real roots exist: positive gives two, zero gives one (a double root), negative gives a complex conjugate pair. The vertex sits at x = −b/2a and is the parabola's minimum when a > 0 or maximum when a < 0.
How to use this calculator
Enter the three coefficients of your quadratic equation written in standard form ax² + bx + c = 0. The leading coefficient a must be non-zero — if a is zero the equation is linear, not quadratic, and the calculator will say so. b and c can be any real numbers including zero and negatives. The calculator returns the roots (two real, one repeated, or a complex conjugate pair depending on the sign of the discriminant), the discriminant itself, the coordinates of the parabola's vertex, the axis of symmetry, and the sum and product of the roots from Vieta's formulas. Units are dimensionless because the formula is purely algebraic.
How the calculation works
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a is derived by completing the square on ax² + bx + c = 0. The expression under the square root, Δ = b² − 4ac, is called the discriminant and determines the nature of the roots. When Δ > 0 the square root is a positive real number and the ± gives two distinct real roots. When Δ = 0 the square root is zero and both branches collapse to a single real root x = −b/(2a), known as a repeated or double root. When Δ < 0 the square root of a negative number is imaginary, producing a complex conjugate pair x = −b/(2a) ± i·√|Δ|/(2a). The vertex of the parabola y = ax² + bx + c lies on its axis of symmetry at x = −b/(2a), with y-coordinate c − b²/(4a); the parabola opens upward when a > 0 and downward when a < 0. Vieta's formulas give two extra checks: the sum of the roots equals −b/a and their product equals c/a.
Worked example
Solve 2x² − 4x − 6 = 0 with a = 2, b = −4, c = −6. The discriminant is (−4)² − 4·2·(−6) = 16 + 48 = 64, which is positive so there are two real roots. The formula gives x = (4 ± √64) / 4 = (4 ± 8) / 4, so x = 3 or x = −1. Check by factoring: 2x² − 4x − 6 = 2(x − 3)(x + 1) ✓. The vertex sits at x = −(−4)/(2·2) = 1, y = −6 − 16/8 = −8, so the minimum of the parabola is the point (1, −8). Vieta: sum = 3 + (−1) = 2 = −b/a = 4/2 ✓; product = 3·(−1) = −3 = c/a = −6/2 ✓. Try x² + 4x + 4 = 0 to see a repeated root (x = −2), or x² + 2x + 5 = 0 to see the complex pair x = −1 ± 2i.
Frequently asked questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a, the closed-form solution to any quadratic equation ax² + bx + c = 0 with a ≠ 0. It is derived by completing the square on the general quadratic and is one of the most widely taught results in school algebra. The ± symbol means the formula generally produces two solutions, which may be two distinct real numbers, the same real number twice, or a pair of complex conjugates depending on the discriminant.
What is the discriminant and what does it tell me?
The discriminant is the expression Δ = b² − 4ac that sits under the square root in the quadratic formula. Its sign classifies the roots without you having to compute them. If Δ > 0 the quadratic has two distinct real roots and the parabola crosses the x-axis at two points. If Δ = 0 the quadratic has one real repeated root and the parabola just touches the x-axis at its vertex. If Δ < 0 the quadratic has no real roots — instead it has a pair of complex conjugate roots and the parabola never crosses the x-axis.
Can a be zero?
No. If a = 0 the equation reduces to bx + c = 0, which is linear and has a single root x = −c/b (assuming b ≠ 0). The quadratic formula divides by 2a, so a = 0 would force a division by zero. The calculator detects this case and asks you to enter a non-zero a.
What are complex or imaginary roots?
When the discriminant is negative the square root in the quadratic formula becomes the square root of a negative number, which is not a real number. Mathematicians extend the reals by defining i = √(−1), and the roots take the form x = −b/(2a) ± i·√|b² − 4ac|/(2a). These are called complex conjugate roots: they have the same real part and opposite imaginary parts. They always come in pairs for any polynomial with real coefficients.
How do I find the vertex of the parabola?
The graph of y = ax² + bx + c is a parabola whose vertex lies at x = −b/(2a). Substitute that x back into the equation to get the vertex y-coordinate, which simplifies to y = c − b²/(4a). The vertex is the minimum of the parabola when a > 0 (it opens upward) and the maximum when a < 0 (it opens downward). The vertical line x = −b/(2a) is the parabola's axis of symmetry: the curve is mirror-symmetric across it.
What are Vieta's formulas?
Vieta's formulas relate the coefficients of a polynomial to symmetric functions of its roots. For ax² + bx + c = 0 with roots r₁ and r₂, the sum r₁ + r₂ = −b/a and the product r₁·r₂ = c/a. They are useful as a quick check on your work: if you compute the roots and they don't satisfy these two relations, something has gone wrong. They also let you build a quadratic from a desired pair of roots: x² − (sum)x + (product) = 0.