AC Method Calculator
Enter any trinomial ax² + bx + c and instantly see the AC product, factor pair table, middle-term split, and fully factored result.
AC Product Analysis
Factor Pair Table
Factored Form
Step-by-Step Solution
What Is the AC Method?
The AC method is a systematic approach for factoring trinomials where the leading coefficient is greater than 1. Instead of guessing factor combinations, you multiply the first coefficient (a) by the last coefficient (c) to form the "AC product." You then search for a pair of numbers whose product equals this AC value and whose sum equals the middle coefficient b.
Once the pair is found, you replace the middle term bx with two terms that use the pair values, creating a four-term polynomial. This four-term expression can then be factored by grouping. For instance, with 6x² + 11x + 3: AC = 18, and the pair (2, 9) satisfies 2 × 9 = 18 and 2 + 9 = 11. Splitting gives 6x² + 2x + 9x + 3 = 2x(3x + 1) + 3(3x + 1) = (3x + 1)(2x + 3).
The AC Method in Three Phases
Compute AC Product
Multiply a × c to get the target product
Find Factor Pair
Locate p, q where p×q = AC and p+q = b
Split & Group
Replace bx with px + qx and factor by grouping
How to Use the AC Method Calculator
Factor any trinomial with a leading coefficient in three steps.
Enter a, b, c
Type the three coefficients from your trinomial ax² + bx + c. For example, for 6x² + 11x + 3, enter a=6, b=11, c=3.
Click "Apply AC Method"
The engine computes a × c, builds a factor pair table, identifies the pair that sums to b, splits the middle term, and applies grouping.
View Full Analysis
See the AC product breakdown, the complete factor pair table with sums highlighted, the step-by-step grouping, and the final factored result.
Worked Examples
Four solved trinomials demonstrating the AC method on various coefficient combinations.
1Factor 6x² + 11x + 3 +−
2Factor 4x² − 8x + 3 +−
3Factor 3x² + 10x + 8 +−
4Factor 5x² − 13x + 6 +−
AC Method Quick Reference
The core steps and conditions for applying the AC method.
| Step | Action | Condition |
|---|---|---|
| 1. Compute AC | Product = a × c | a ≠ 0 |
| 2. Find pair | p × q = AC, p + q = b | Integer pair must exist |
| 3. Split bx | bx → px + qx | Creates 4 terms |
| 4. Group & factor | Factor by grouping | Shared binomial required |
Frequently Asked Questions
Common questions about the AC method for factoring trinomials.
What is the AC method in factoring? +−
The AC method multiplies the leading coefficient a by the constant term c, then finds two numbers whose product equals a×c and whose sum equals b. These numbers are used to split the middle term and factor by grouping.
When do I need the AC method instead of simple factoring? +−
Use the AC method whenever the leading coefficient a is not 1. Simple trinomial factoring (finding p and q where p+q = b and p×q = c) only works directly when a = 1.
What if no factor pair sums to b? +−
If no pair of integers whose product is a×c sums to b, the trinomial is irreducible over the integers and cannot be factored using this method.
Is the AC method the same as the grouping method? +−
The AC method is a two-part process: first, find the pair that splits the middle term, then apply factoring by grouping. Grouping is a general technique; the AC method specifically uses it on trinomials.
Can the AC method produce fractional factors? +−
No — the AC method exclusively works with integer factor pairs, so the resulting binomial factors always have integer coefficients.
How do I verify my AC method result? +−
Multiply the two binomial factors using FOIL and check that the expanded form matches the original trinomial ax² + bx + c.
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