When the coefficient of x² is 1, factoring reduces to a number puzzle: find two numbers whose product is the constant term and whose sum is the middle coefficient.
The sign pattern of the trinomial tells you the signs of the two numbers before you start listing factor pairs, which cuts the search in half.
How to solve it, step by step
Identify b and c
In x² − 5x − 24, b = −5 and c = −24.
Read the signs
If c is positive, both numbers share the sign of b. If c is negative, the numbers have opposite signs and the larger absolute value takes the sign of b.
List factor pairs of |c|
Pairs for 24: 1·24, 2·12, 3·8, 4·6.
Find the pair with the right sum (or difference)
Opposite signs, so look for a difference of 5: 3 and 8. The larger takes b's sign: −8 and +3. Check: −8 + 3 = −5 ✓ and (−8)(3) = −24 ✓.
Write the factors and verify by expanding
(x − 8)(x + 3) = x² + 3x − 8x − 24 = x² − 5x − 24 ✓.
Worked example
Factor x² + 11x + 28.
- b = 11, c = 28 → both numbers positive
- Pairs of 28: 1·28, 2·14, 4·7
- 4 + 7 = 11 ✓
Answer (x + 4)(x + 7)
Factor x² − 2x − 35.
- c < 0 → opposite signs; difference 2
- Pairs of 35: 1·35, 5·7 → 7 − 5 = 2
- Larger takes b's sign: −7 and +5
Answer (x − 7)(x + 5)
Mistakes DeltaMath marks wrong
- Choosing a pair with the right product but the wrong sum, then not expanding to check.
- Getting the sign backwards on the larger number when c is negative.
- Forgetting that x² + 0x − 16 is a difference of squares: (x − 4)(x + 4).
- Writing (x + 4)(x + 7) as x + 4 · x + 7 without parentheses in the answer box.
How to type the answer
- Type both factors with parentheses: (x+4)(x+7). Order does not matter.
- If the trinomial cannot be factored over the integers, choose "prime" or "not factorable" when offered.