A square with two letters and numbers
A square tile has side 8a + 3b. Here a and b are lengths. How much floor does it cover?
- A square of side 8a + 3b.
- Two corner squares: 64a² and 9b².
- Two crossings of 24ab each.
- Add all four blocks.
The rule
The tile covers 64a² + 48ab + 9b².
Rule: Square each part. The middle is twice their product, with the sign between the parts.
A part can have a number in front, like 8a. Square the number and the letter: (8a)² = 8² × a² = 64a². Likewise (3b)² = 9b². Now the middle: 2 × 8a × 3b = 48ab. It is twice the product of 8a and 3b. So (8a + 3b)² = 64a² + 48ab + 9b².
With a minus, the middle takes the minus sign. Here (5m)² = 25m² and (4n)² = 16n². The middle is 2 × 5m × 4n = 40mn. So (5m − 4n)² = 25m² − 40mn + 16n².
Reading it backwards
Is 16p² + 40pq + 25q² a square? Its ends, the first and last terms, are (4p)² and (5q)². Twice the product of 4p and 5q is 2 × 4p × 5q = 40pq. That matches the middle. So it is (4p + 5q)².
| Line | Ends | Twice the product | Verdict |
|---|---|---|---|
16p² + 40pq + 25q² | (4p)² and (5q)² | 40, and it reads 40 | (4p + 5q)² |
25p² − 40pq + 16q² | (5p)² and (4q)² | 40, and it reads −40 | (5p − 4q)² |
16p² + 20pq + 25q² | (4p)² and (5q)² | 40, but it reads 20 | not a square |
12x² + 36xy + 27y² | 12 and 27 are not squares | take out 3 first | 3(2x + 3y)² |
Sometimes the terms share a common factor. A common factor divides every term. Take it out first. 12x² + 36xy + 27y² is 3(4x² + 12xy + 9y²), and the bracket is (2x + 3y)².
Read the line. Tap the bracket that gives it.
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