Splitting the middle term
A garden has area x² + 9x + 20 square metres. Here x is a length in metres. No sides are given. What are they? Pick the pair that splits it.
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- A rectangle, sides x + 4 and x + 5. Here x = 6.
- Corner: x² = 36.
- Top strip: 5x = 30.
- Side strip: 4x = 24.
- Corner: 4 × 5 = 20.
- Total: 36 + 30 + 24 + 20 = 110. So x² + 9x + 20 = (x + 4)(x + 5).
The rule
The sides are x + 4 and x + 5.
Look for two numbers, p and q. They are the pair. The pair does two jobs at once. It adds to the middle number, 9. It multiplies to the last number, 20. A pair that does only one job is wrong.
(x + p)(x + q) = x² + (p + q)x + pq
| Pair | Adds to | Multiplies to | Both jobs? |
|---|---|---|---|
| 1 and 20 | 21 | 20 | no |
| 2 and 10 | 12 | 20 | no |
| 3 and 6 | 9 | 18 | no |
| 4 and 5 | 9 | 20 | yes |
| −4 and −5 | −9 | 20 | no |
So the garden is (x + 4)(x + 5). At x = 2 the area is 4 + 18 + 20 = 42. The brackets give 6 × 7 = 42.
A negative last number
x² + x − 12 needs a pair that adds to 1 and multiplies to −12. Try 4 and −3. Now 4 + (−3) = 1 and 4 × (−3) = −12. So it is (x + 4)(x − 3).
When no pair exists
x² + x + 1 needs a pair that adds to 1 and multiplies to 1. We want whole numbers. Pairs with product 1 are 1 and 1, or −1 and −1. Neither adds to 1. So it does not split into whole numbers.