Quadratic Factoring Calculator
Enter any quadratic equation and instantly get its factored form, roots, vertex, discriminant analysis, and an interactive parabola graph.
Discriminant Analysis
Factored Form & Roots
Vertex & Properties
Parabola Graph
What Is Quadratic Factoring?
Quadratic factoring transforms a second-degree polynomial ax² + bx + c into a product of two linear binomials. When you factor x² − 5x + 6, you discover that it equals (x − 2)(x − 3), which immediately reveals the roots x = 2 and x = 3 — the points where the parabola crosses the x-axis.
The discriminant Δ = b² − 4ac is the key to understanding any quadratic. A positive discriminant means two distinct real roots. Zero means one repeated root (the parabola just touches the x-axis). A negative discriminant means no real roots at all — the parabola floats entirely above or below the x-axis.
Discriminant Decision Chart
Two Real Roots
Parabola crosses x-axis twice
One Repeated Root
Parabola touches x-axis once
No Real Roots
Parabola does not cross x-axis
How to Use the Quadratic Factoring Calculator
Solve any quadratic equation in three quick steps.
Enter a, b, c
Type the three coefficients from your quadratic equation ax² + bx + c = 0. Use negative signs for subtraction.
Click "Solve Quadratic"
The engine computes the discriminant, determines the root type, and factors the expression if possible.
View Full Solution
See the factored form, exact roots, vertex coordinates, parabola direction, and an interactive graph.
Worked Examples
Four solved quadratic equations demonstrating each discriminant case and factoring technique.
1x² − 5x + 6 = 0 +−
2x² + 4x + 4 = 0 +−
3x² − 25 = 0 +−
42x² + 3x − 2 = 0 +−
Quadratic Formulas Quick Reference
Essential formulas for working with quadratic equations.
| Formula | Expression | Purpose |
|---|---|---|
| Quadratic Formula | x = (−b ± √Δ) / 2a | Find exact roots |
| Discriminant | Δ = b² − 4ac | Determine root type |
| Vertex x-coord | h = −b / (2a) | Find axis of symmetry |
| Vertex Form | a(x − h)² + k | Identify vertex (h, k) |
| Sum of Roots | r₁ + r₂ = −b/a | Vieta's formula |
| Product of Roots | r₁ × r₂ = c/a | Vieta's formula |
Frequently Asked Questions
Common questions about quadratic equations and factoring.
What is a quadratic equation? +−
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0, producing a parabola when graphed and having at most two real roots.
How does the discriminant determine factorability? +−
If Δ = b² − 4ac is a perfect square ≥ 0, the quadratic factors over the integers; if Δ < 0, it has no real roots.
What is the quadratic formula? +−
x = (−b ± √Δ) / 2a gives the exact roots of any quadratic, whether or not it factors neatly over the integers.
Can this calculator handle negative leading coefficients? +−
Yes — enter any integer value for a, b, and c including negatives, and the calculator factors or reports irreducibility accordingly.
What does the vertex of a parabola represent? +−
The vertex is the minimum or maximum point of the parabola, located at x = −b/(2a) with y-value f(−b/(2a)).
When should I use completing the square instead? +−
Completing the square works for any quadratic and is preferred when deriving vertex form or when the discriminant is not a perfect square.
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