The numbers are 21 and 22.

Let n represent one of the numbers. Since the numbers are consecutive then let n + 1 represent the other number.

The difference of their squares equals 43. So we just need to solve the following equation.
(n +1)^2 - n^2 = 43
(n^2 + 2n +1) - n^2 = 43
2n + 1 = 43
2n = 42
n = 21

Now we can ckeck our answer.
22^2 - 21^2 = 43
484 - 441 = 43
43 = 43

 

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