Question

Which shows a difference of squares?

10 y squared minus 4 x squared
16 y squared minus x squared
8 x squared minus 40 x + 25
64 x squared minus 48 x + 9

Answer 1

Only 16y^2 − x^2 is a (nontrivial) difference of squares:

16y^2 − x^2 = (4y)^2 − (x)^2 = (4y − x)(4y + x).

Notes on the others:
- 10y^2 − 4x^2 = 2(5y^2 − 2x^2) is not a difference of two perfect squares with rational coefficients (it would be (√10 y − 2x)(√10 y + 2x) with irrationals).
- 8x^2 − 40x + 25 is not a difference of squares (its discriminant = 800, not a perfect square).
- 64x^2 − 48x + 9 = (8x − 3)^2, a perfect square, not a (nontrivial) difference of squares.