Proposing one of your own
You write (a + b)² − (a − b)² = 4ab on a page. You made it up. Is it true? First try three other claims.
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(a + b)² − (a − b)² = 4ab. Step 1 of 6
- Claim:
(a + b)² − (a − b)² = 4ab. - At a = 7, b = 3: both sides give 84.
- Opening both sides: every term matches.
- A second claim:
(x + 3)² = x² + 9. - At x = 0: both sides give 9.
- At x = 1: 16 and 10 disagree.
The rule
Yes. Your claim is true for any numbers a and b.
An identity is an equation that is true for every value. An equation may be true for only some values.
One disagreement settles it: the claim is not an identity. Agreement at a few values does not settle it. Opening both sides out does. Opening out means multiplying out the brackets.
Open the claim above. (a + b)² is a² + 2ab + b². (a − b)² is a² − 2ab + b². Subtract the second from the first. The a² terms cancel, and so do the b² terms. The middle terms give 2ab − (−2ab) = 4ab. That works for any a and b. So the claim is an identity.
| Claim | Agrees at | Fails at |
|---|---|---|
x² + 2 = 3x | x = 1, x = 2 | every other value |
x³ = x | x = −1, 0, 1 | x = 2, where 8 meets 2 |
(x + 3)² = x² + 9 | x = 0 | x = 1, where 16 meets 10 |