What you need
- Quadratic expression (e.g., x² + 5x + 6)
- Paper and pencil
- List of factor pairs
- Basic understanding of multiplication and addition
Every step
- 1
Check that this method fits your quadratic
This method applies to quadratics whose x² coefficient is 1, such as x² + 5x + 6. Identify b = 5 and c = 6. You need two integers that multiply to c and add to b. If the leading coefficient is not 1, or no integer pair works, use an appropriate different method rather than forcing these factors.
CHECKPOINT · Your expression is x² + bx + c with leading coefficient 1. - 2
List factor pairs of the constant term
Find pairs of integers that multiply to the constant term (c). For x² + 5x + 6, c = 6. The pairs are (1,6) and (2,3). Determine the sum of each pair.
CHECKPOINT · Several factor pairs for c are listed. - 3
Select the pair that sums to the linear coefficient
From the previous step, identify the pair whose sum equals b. For x² + 5x + 6, (2,3) sum to 5. This is the pair to use for factoring.
CHECKPOINT · Chosen pair correctly sums to b (e.g., 2 + 3 = 5). - 4
Write the factored form
Rewrite as (x + m)(x + n), where m and n are your chosen numbers. For our example, (x + 2)(x + 3). Expand to check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
CHECKPOINT · Factored form multiplies back to original quadratic. - 5
Handle negative constants
If the constant term is negative, list positive and negative factor pairs. For x² – x – 6: factor pairs for -6 are (-1,6), (1,-6), (2,-3), (-2,3). Pick the pair summing to -1: (2,-3). So, factors: (x + 2)(x – 3).
CHECKPOINT · Factored form checks against original, including correct signs.
Common mistakes
- Confusing addition and subtraction with negative constants leads to sign errors.
- Not every quadratic can be factored into rationals; use quadratic formula when needed.
Finished when
Expression rewritten as the product of two binomials, verified by expansion.
Questions about this task
What if I can’t find integers that work?
If no pair of integers multiply to c and add to b, the quadratic can’t be factored with integers. Use the quadratic formula or complete the square to find solutions.
Does factoring work for all quadratics?
Factoring only works easily when the quadratic can be written with integer factors. Otherwise, use alternative methods like the quadratic formula.
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