The easy way round an awkward product
You can find 98 × 98 in your head. It needs one idea.
98 = 100 − 2. Step 1 of 4
98 = 100 − 2.(100 − 2)² = 100² − 2×100×2 + 2².= 10,000 − 400 + 4.= 9,604.
Identity — a line that is true at every value you put in.
| Name | Identity | A check, both sides |
|---|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² | a = 5, b = 2: 49 |
| Square of a difference | (a − b)² = a² − 2ab + b² | a = 10, b = 3: 49 |
| Difference of squares | (a + b)(a − b) = a² − b² | a = 12, b = 5: 119 |
(a + b)(a − b) gives a² − ab + ab − b². The middle terms add to 0.
A round number ends in 0, such as 50 or 100.
Rule: write each number near a round one. Then pick the pattern that fits.
47 × 53 is (50 − 3)(50 + 3), as 47 = 50 − 3 and 53 = 50 + 3. These two are partners: each is 3 from 50, on opposite sides. The third pattern gives 50² − 3², which is 2,500 − 9 = 2,491.
Watch out!
Reading 98 × 98 as (100 − 2)(100 + 2) gives 9,996. Both numbers are 98, so both brackets must be (100 − 2). Use the square of a difference: 9,604.
Slide a number. See its bar and the pattern that fits.
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