If the difference of two numbers is 8 and the difference of their squares is…

Concept: For any two numbers, the difference of their squares equals the product of their sum and their difference — a2 − b2 = (a + b)(a − b). This identity lets us find the sum of two numbers directly when their difference and the difference of their squares are both known, without solving a quadratic equation.

Application:

1. Let the two numbers be X and Y, with X > Y.

2. Given: X − Y = 8.

3. Given: X2 − Y2 = 160.

4. By the identity, X2 − Y2 = (X + Y)(X − Y), so (X + Y)(X − Y) = 160.

5. Substituting X − Y = 8: (X + Y)(8) = 160, so X + Y = 20.

6. Solving X − Y = 8 and X + Y = 20 together: adding the two equations gives 2X = 28, so X = 14; then Y = X − 8 = 6.

Cross-check: The difference 14 − 6 = 8 matches the given difference, and the difference of squares 142 − 62 = 196 − 36 = 160 matches the given value. So the numbers are 6 and 14.