Calculator Inputs
Enter coefficients for a trinomial in the form ax2 + bx + c.
Example Data Table
| Trinomial | Expected Middle Test | Discriminant | Classification | Factored Form |
|---|---|---|---|---|
| x² + 6x + 9 | ±2√(1×9) = ±6 | 0 | Perfect square | (x + 3)² |
| 4x² + 12x + 9 | ±2√(4×9) = ±12 | 0 | Perfect square | (2x + 3)² |
| 9x² - 12x + 4 | ±2√(9×4) = ±12 | 0 | Perfect square | (3x - 2)² |
| x² + 5x + 6 | ±2√(1×6) ≈ ±4.899 | 1 | Not perfect square | (x + 2)(x + 3) |
Formula Used
Positive middle term identity: (mx + n)² = m²x2 + 2mnx + n²
Negative middle term identity: (mx - n)² = m²x2 - 2mnx + n²
Middle term check: For a perfect square trinomial, the middle coefficient must equal ±2√(ac) when the first and last coefficients are non-negative.
Discriminant test: D = b² - 4ac. A single-variable quadratic is a repeated square binomial when D = 0 and a > 0.
Repeated root: x = -b / 2a
How to Use This Calculator
- Enter the leading coefficient, middle coefficient, and constant term.
- Pick the variable symbol you want displayed in the expression.
- Set decimal precision for roots, square roots, and graph labels.
- Choose a graph half-width to inspect behavior near the vertex.
- Click Analyze Trinomial to generate the result above the form.
- Review the factorization, discriminant, repeated root, and complete-square form.
- Inspect the plot to see whether the curve touches the axis once.
- Use the CSV and PDF buttons to save the current analysis.
FAQs
1. What is a perfect square trinomial?
It is a quadratic expression produced by squaring a binomial. Common forms are a² + 2ab + b² and a² - 2ab + b².
2. Why does the middle term matter so much?
The middle term must exactly match ±2ab, or equivalently ±2√(ac) in coefficient form. If it does not match, the trinomial is not a perfect square.
3. Can decimals still form a perfect square trinomial?
Yes. A trinomial can still be a perfect square over real numbers when the discriminant is zero, even if the visible coefficients are decimals.
4. What does a zero discriminant tell me?
It shows the quadratic has one repeated root. That means the parabola touches the horizontal axis once and the expression can be written as a squared linear factor.
5. Why does the graph touch the axis only once?
A squared binomial gives a repeated root. Geometrically, the parabola reaches the axis at its vertex and turns back without crossing.
6. Is every quadratic with D = 0 a perfect square trinomial?
Over real numbers with a positive leading coefficient, yes. It can be written as a squared linear expression, though not always with integer coefficients.
7. What is the difference between integer and real perfect squares?
An integer perfect square trinomial uses integer square roots in the factor form. A real perfect square may require irrational or decimal coefficients.
8. When should I use completing the square instead?
Use completing the square when the trinomial is not already an obvious perfect square. It rewrites the quadratic into vertex form and shows how far it is from a square.