Calculator
Example Data Table
| Quadratic Expression | a | b | c | Factors | Roots |
|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 | (x - 2)(x - 3) | 2, 3 |
| 2x² + 5x + 2 = 0 | 2 | 5 | 2 | (2x + 1)(x + 2) | -1/2, -2 |
| x² + 4x + 4 = 0 | 1 | 4 | 4 | (x + 2)(x + 2) | -2, -2 |
| x² + x - 6 = 0 | 1 | 1 | -6 | (x + 3)(x - 2) | -3, 2 |
Formula Used
This calculator first checks the discriminant, then looks for a useful split pair for integer factoring. When the expression factors into two linear terms, each factor is set equal to zero to obtain the roots.
It also reports the axis of symmetry, vertex, y-intercept, and a graph of the parabola to support checking.
How to Use This Calculator
- Enter the coefficients a, b, and c from the quadratic equation.
- Keep the equation in standard form ax² + bx + c = 0.
- Click Find Roots to process the expression.
- Read the roots, factor form, and factoring status above the form.
- Review the graph to see where the parabola crosses the x-axis.
- Download the result as CSV or PDF when needed.
Frequently Asked Questions
1. What does factoring mean in this calculator?
Factoring rewrites the quadratic as a product of two linear expressions. After that, each factor is set equal to zero to find the roots.
2. Can this solve every quadratic equation?
It analyzes every valid quadratic. Some expressions factor neatly, while others only produce irrational or non-real roots. The calculator clearly reports that status.
3. Why do I need values for a, b, and c?
Those coefficients define the equation in standard form. The calculator uses them to check factor pairs, compute roots, and draw the parabola.
4. What happens if a equals zero?
Then the expression is not quadratic. This tool requires a nonzero leading coefficient because factoring and graphing depend on the parabola form.
5. What if the discriminant is negative?
A negative discriminant means there are no real roots. The graph will not cross the x-axis, and real linear factorization is not available.
6. Why is the graph useful?
The graph shows the parabola, the x-intercepts, and the turning point. It helps you visually confirm whether the computed roots make sense.
7. Why would I download CSV or PDF?
Exports help with homework records, classroom examples, revision notes, or sharing results with students and colleagues in a simple format.
8. How can I verify the roots manually?
Substitute each root back into ax² + bx + c. A correct root makes the expression equal zero, apart from tiny rounding differences.