To determine which of the given expressions represent a difference of squares, we must understand the form of a difference of squares. A difference of squares is an expression in the form [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a + b)(a - b)\)[/tex].
Let's analyze each of the given expressions:
1. [tex]\(10 y^2 - 4 x^2\)[/tex]:
- We can rewrite this expression as [tex]\(2(5y^2) - 4x^2 = 2(5y^2) - 2(2x)^2\)[/tex].
- Simplifying further, it becomes just [tex]\(2((\sqrt{5}y)^2 - 2x^2)\)[/tex].
- This expression doesn't fit directly into the difference of squares form since there is a common factor of 2.
2. [tex]\(16 y^2 - x^2\)[/tex]:
- Rewrite it as [tex]\((4y)^2 - x^2\)[/tex].
- This is in the form [tex]\(a^2 - b^2\)[/tex] with [tex]\(a = 4y\)[/tex] and [tex]\(b = x\)[/tex].
- Thus, this can be factored as [tex]\((4y + x)(4y - x)\)[/tex].
- Therefore, [tex]\(16 y^2 - x^2\)[/tex] is indeed a difference of squares.
3. [tex]\(8 x^2 - 40 x + 25\)[/tex]:
- Rewrite this in the form of a perfect square trinomial:
[tex]\(8x^2 - 40x + 25 = (2\sqrt{2}x)^2 - (2\sqrt{2}x \times 2.5 \sqrt{2}x) + (2.5\sqrt{2}x)^2 = (2\sqrt{2}x - 2.5\sqrt{2}x)^2 = 8(x - \frac{5}{2})^2\)[/tex].
- This is not in the form [tex]\(a^2 - b^2\)[/tex] but rather a perfect square trinomial.
4. [tex]\(64 x^2 - 48 x + 9\)[/tex]:
- Rewrite this in the form of a perfect square trinomial:
[tex]\(64x^2 - 48x + 9 = (8x)^2 - (6x)(6x) + (6)^2 = (8x - 3)^2\)[/tex].
- This is also not in the form [tex]\(a^2 - b^2\)[/tex] but rather a perfect square trinomial.
Therefore, out of the given expressions, the only one that shows a difference of squares is [tex]\(16 y^2 - x^2\)[/tex].
