| Statement | Comment: |
| -20 = -20 | Hard to argue with! |
| 16 - 36 = 25 - 45 | Rewriting |
| 16 - 36 + 81/4 = 25 - 45 + 81/4 | Add 81/4 to each side |
| (4 - 9/2)2 = (5 - 9/2)2 | Rewriting |
| 4 - 9/2 = 5 - 9/2 | Sqrt of each side |
| 4 = 5 | Add 9/2 to each side |
| 1 = 2 | Subtract 3 from each side |
| Statement | Comment: |
| 16 = 16 | Reflexive property of "=" |
| 4 + 12 = 16 | From definition of "+" |
| 4 - 12 = 16 - 24 | Subtract 24 from both sides |
| 4 - 12 + 9 = 16 - 24 + 9 | Add 9 to both sides, completing squares |
| (2 - 3)2 = (4 - 3)2 | Factor, rewriting as squares |
| 2 - 3 = 4 - 3 | Sqrt of each side |
| 2 = 4 | Add 3 to each side |
| 1 = 2 | Divide by 2 |
Exposing the error: one side squares a negative number and the other squares its additive inverse (I think that's correct term, it's been 30 years). That's equivalent to saying that 3 = -3 because 32 = (-3)2.
Here's a different one, with a different flaw:
x = y
x2 = xy
x2 - y2 = xy - y2
(x - y)(x + y) = y(x - y)
(x - y)(x + y)/(x - y) = y(x - y)/(x - y)
x + y = y
y + y = y
2y = y
2 = 1
The flaw is this:
x = y
x - y = 0
(x - y)(x + y)/(x - y) is division by zero.