Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Handles real distinct, real equal, and complex roots with step-by-step solutions. See also Slope Calculator and Exponent Calculator.
Enter coefficients for ax² + bx + c = 0
How to Use the Quadratic Formula
The quadratic formula solves any equation of the form ax² + bx + c = 0, where a ≠ 0. Enter the three coefficients (a, b, c) and the calculator finds the roots. The discriminant (b² − 4ac) determines the nature of the roots: positive means two distinct real roots, zero means one repeated root, and negative means two complex conjugate roots. The calculator also provides the vertex, axis of symmetry, and a complete step-by-step solution.
Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Discriminant:
Δ = b² − 4ac
If Δ > 0: two distinct real roots
If Δ = 0: one repeated real root
If Δ < 0: two complex conjugate roots
Vertex:
x = −b / 2a, y = f(−b / 2a)
Example Calculation
Solve: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
Δ > 0 → Two distinct real roots
x₁ = (5 + √1) / 2 = 6/2 = 3
x₂ = (5 − √1) / 2 = 4/2 = 2
Roots: x = 2 and x = 3
Vertex: (2.5, −0.25)
Quadratic Equation Reference Table
| Equation | Δ | Roots | Nature |
|---|---|---|---|
| x² − 5x + 6 = 0 | 1 | x = 2, x = 3 | Real distinct |
| x² − 4x + 4 = 0 | 0 | x = 2 (repeated) | Real equal |
| x² + 1 = 0 | −4 | x = ±i | Complex |
| 2x² − 7x + 3 = 0 | 25 | x = 3, x = 0.5 | Real distinct |
| x² − 2x + 5 = 0 | −16 | x = 1 ± 2i | Complex |
| x² − 6x + 9 = 0 | 0 | x = 3 (repeated) | Real equal |
| 3x² + 2x − 1 = 0 | 16 | x = 1/3, x = −1 | Real distinct |
Frequently Asked Questions
What if a = 0?
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The solution is simply x = −c/b. The quadratic formula requires a ≠ 0.
What are complex roots?
When the discriminant is negative, the square root produces an imaginary number. Complex roots come in conjugate pairs: a + bi and a − bi, where i = √(−1). These roots mean the parabola doesn't cross the x-axis.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola. For a > 0, it's the minimum point; for a < 0, it's the maximum. The x-coordinate of the vertex is −b/(2a), and the y-coordinate is found by substituting back into the equation.
Can I solve quadratics by factoring instead?
Yes, when the roots are rational numbers. For example, x² − 5x + 6 = (x − 2)(x − 3) = 0. However, the quadratic formula works for all quadratic equations, including those that can't be easily factored.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = −b/(2a) that divides the parabola into two mirror-image halves. The vertex lies on this line, and the two roots (if real) are equidistant from it.