One square minus another
Two square plots have edges 101 and 99 paces. What is the gap between their areas, without squaring either number? Try the lab.
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- A square of side a = 5, with a b-by-b corner marked off (b = 3).
- What is left has area a² − b² = 16.
- Cut it into two rectangles: a(a − b) = 10 and b(a − b) = 6.
- Slide them together: one rectangle, (a + b) wide and (a − b) tall. 8 × 2 = 16.
- So a² − b² = (a − b)(a + b).
The rule
The area gap is 400 square paces.
a² − b² = (a − b)(a + b). Here a and b are the two edges. The edge a is longer. The first bracket is the gap between the edges. The second is their total.
The gap is 101 − 99 = 2. The total is 101 + 99 = 200. The area gap is 2 × 200 = 400 square paces.
Numbers just below and above 100
98 × 102 is (100 − 2)(100 + 2). Here a = 100 and b = 2. So it is 100² − 2² = 10000 − 4 = 9996.
Reading it backwards
To factor is to write as brackets multiplied. Here x stands for any number.
x² − 9 is x² minus 3². So it factors as (x − 3)(x + 3). Check at x = 4: both give 7.
Both parts must be squares. 4x² − 25 is (2x)² − 5², so it is (2x − 5)(2x + 5). But 7 is not the square of a whole number. So x² − 7 does not factor with whole numbers.
| Expression | Factors | At x = 2 |
|---|---|---|
x² − 4 | (x − 2)(x + 2) | 0 and 0 |
9x² − 1 | (3x − 1)(3x + 1) | 35 and 35 |
x² + 4 | none | a sum, not a gap |
Watch out!
a² − b² is not (a − b)². Try a = 5 and b = 3. The first is 16. The second is 4.