Quadratic Formula Calculator — Solve ax² + bx + c = 0
Solve any quadratic equation with the quadratic formula. Find both roots, discriminant, vertex, axis of symmetry, and factored form with step-by-step solutions.
Factoring Trinomials Calculator
Factor ax² + bx + c into linear factors. Find roots, discriminant, vertex, and see the AC method step by step with a factor-pair analysis table.
Factor ax² + bx + c — find roots, discriminant, AC method steps, and factor pairs
Trinomial⧉
x² + 5x + 6
a=1, b=5, c=6
Factored Form⧉
(x + 2)(x + 3)
Product of linear factors (if factorable over ℝ)
Discriminant (Δ)⧉
1.00
b² − 4ac = 5² − 4(1)(6)
Root Type⧉
Two distinct real roots
Δ > 0
Root 1⧉
-2.000000
x₁ = (−b + √Δ) / 2a
Root 2⧉
-3.000000
x₂ = (−b − √Δ) / 2a
Vertex⧉
(-2.5000, -0.2500)
Minimum/maximum point of parabola
AC Product⧉
6.00
a × c = 1 × 6
AC Method Steps
Step 1: Compute a × c = 1 × 6 = 6
Step 2: Find two numbers m, n such that m × n = 6 and m + n = 5
Step 3: m = -1, n = 6 ✓
Step 4: Rewrite: 1x² + -1x + 6x + 6
Step 5: Factor by grouping → (x + 2)(x + 3)
Factor Pairs of AC = 6
| m | n | m × n | m + n | Match b? |
|---|---|---|---|---|
| -6 | -1 | 6 | -7 | ✗ |
| -6 | 1 | -6 | -5 | ✗ |
| -3 | -2 | 6 | -5 | ✗ |
| -3 | 2 | -6 | -1 | ✗ |
| -2 | -3 | 6 | -5 | ✗ |
| -2 | 3 | -6 | 1 | ✗ |
| -1 | -6 | 6 | -7 | ✗ |
| -1 | 6 | -6 | 5 | ✓ Yes |
| 1 | 6 | 6 | 7 | ✗ |
| 1 | -6 | -6 | -5 | ✗ |
| 2 | 3 | 6 | 5 | ✓ Yes |
| 2 | -3 | -6 | -1 | ✗ |
| 3 | 2 | 6 | 5 | ✓ Yes |
| 3 | -2 | -6 | 1 | ✗ |
| 6 | 1 | 6 | 7 | ✗ |
| 6 | -1 | -6 | 5 | ✓ Yes |
Discriminant Classification
Δ < 0 (Complex)
Δ = 0 (Repeated)
Δ > 0 (Distinct)
Planning notes, formulas, and examples
About the Factoring Trinomials Calculator
Factoring trinomials is one of the most fundamental skills in algebra. A trinomial of the form ax² + bx + c can often be written as a product of two linear factors, revealing its roots — the x values where the expression equals zero. This is essential for solving quadratic equations, simplifying rational expressions, graphing parabolas, and tackling higher-level algebra and calculus problems. The most systematic approach for factoring trinomials is the AC method (also called factoring by grouping). You multiply a and c, then search for two numbers whose product equals ac and whose sum equals b. If such a pair exists, you can split the middle term and factor by grouping. This page lays out that full workflow: enter a, b, and c and review the factored form, both roots (real or complex), the discriminant Δ = b² − 4ac, the vertex of the parabola, and a table of factor pairs of the AC product with the matching pair highlighted. If the trinomial cannot be factored over the integers, the calculator still provides decimal roots via the quadratic formula. Eight presets cover classic examples — simple trinomials like x² + 5x + 6, general trinomials like 2x² + 7x + 3, difference of squares like x² − 9, and perfect square trinomials like 4x² − 4x + 1. A color-coded discriminant classifier shows at a glance whether the roots are real and distinct, repeated, or complex. Whether you are learning to factor in Algebra 1, reviewing for a standardized test, or teaching factoring techniques, the page gives you the step sequence as well as the final answer.
When This Page Helps
Factoring trinomials is usually less about getting any root and more about seeing whether the expression really breaks into clean linear factors. This page is useful because it keeps the original trinomial, factored form, discriminant, and root type together, so you can tell whether the AC pair worked, whether a perfect-square pattern appeared, or whether the quadratic formula is the only path left.
How to Use the Inputs
- Enter Coefficient a and Coefficient b in the input fields.
- Select the mode, method, or precision options that match your factoring trinomials problem.
- Read Trinomial first, then use Factored Form to confirm your setup is correct.
- Try a preset such as "x²+5x+6" to test a known case quickly.
Formula used
ax² + bx + c = a(x − r₁)(x − r₂)
Discriminant: Δ = b² − 4ac
Roots: x = (−b ± √Δ) / (2a)
AC method: find m, n where m·n = ac and m+n = b
Vertex: (−b/(2a), f(−b/(2a)))Example Calculation
Result: Trinomial shown by the calculator
Using the preset "x²+5x+6", the calculator evaluates the factoring trinomials setup, applies the selected algebra rules, and reports Trinomial with supporting checks so you can verify each transformation.
Tips & Best Practices
- Always check for a GCF first — factor it out before applying the AC method.
- If the discriminant is negative, the trinomial cannot be factored over the reals.
- A perfect-square trinomial has Δ = 0 and factors as (px + q)².
- Difference of squares (b = 0, c < 0) factors as (√a·x + √|c|)(√a·x − √|c|).
- Practice recognizing patterns — many textbook trinomials use small integer roots.
How This Factoring Trinomials Calculator Works
This calculator takes Coefficient a, Coefficient b, Coefficient c and applies the relevant factoring trinomials relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Interpreting Results
Start with the primary output, then use Trinomial, Factored Form, Discriminant (Δ), Root Type to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
Study Strategy
Sources & Methodology
Last updated:
Frequently Asked Questions
The AC method multiplies a and c, then finds two numbers m and n whose product is ac and whose sum is b. You then rewrite bx as mx + nx and factor by grouping.
If no integer pair m, n satisfies m·n = ac and m+n = b, the trinomial is "prime" over the integers. You can still find roots using the quadratic formula — the roots will be irrational or complex.
The discriminant Δ = b²−4ac determines the root type: Δ > 0 means two distinct real roots, Δ = 0 means one repeated root, and Δ < 0 means two complex conjugate roots.
Use the AC method: multiply a×c, find the pair summing to b, split the middle term, and factor by grouping. This works for any trinomial regardless of the leading coefficient.
A perfect square trinomial has the form a²x² ± 2abx + b² and factors as (ax ± b)². Its discriminant is exactly zero.
This calculator is designed for degree-2 trinomials (ax² + bx + c). For higher degrees, try polynomial division or use a dedicated polynomial factoring calculator.
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