To determine which expressions can be written as a difference of squares, we need to check if they can be expressed in the form [tex]\(A^2 - B^2\)[/tex]. Let's analyze each of the given expressions:
### Expression 1: [tex]\(10 y^2 - 4 x^2\)[/tex]
This can be written as:
[tex]\[ 10 y^2 - 4 x^2 = 10 y^2 - (2x)^2 \][/tex]
We can see that:
- [tex]\(A = \sqrt{10} y\)[/tex]
- [tex]\(B = 2x\)[/tex]
Thus, this represents a difference of squares:
[tex]\[ (\sqrt{10}y)^2 - (2x)^2 \][/tex]
### Expression 2: [tex]\(16 y^2 - x^2\)[/tex]
This can be written as:
[tex]\[ 16 y^2 - x^2 = (4y)^2 - x^2 \][/tex]
We can see that:
- [tex]\(A = 4y\)[/tex]
- [tex]\(B = x\)[/tex]
Thus, this represents a difference of squares:
[tex]\[ (4y)^2 - x^2 \][/tex]
### Expression 3: [tex]\(8 x^2 - 40 x + 25\)[/tex]
This can be written as:
[tex]\[ 8 x^2 - 40 x + 25 \][/tex]
To see if it can be simplified into a difference of squares, note the middle term (linear term, [tex]\( -40x \)[/tex]) and the constant term (25). This expression seems to be a quadratic trinomial. To determine if it's a perfect square, let's check if it can be expressed as [tex]\((Ax + B)^2\)[/tex]:
[tex]\[ 8 x^2 - 40 x + 25 \][/tex]
So, this expression does not fit into the form [tex]\(A^2 - B^2\)[/tex].
### Expression 4: [tex]\(64 x^2 - 48 x + 9\)[/tex]
This can be written as:
[tex]\[ 64 x^2 - 48 x + 9 \][/tex]
Again, to confirm if it's a perfect square or fits the form of difference of squares [tex]\(A^2 - B^2\)[/tex], we check:
[tex]\[ 64 x^2 - 48 x + 9 \][/tex]
This also does not fit the form [tex]\(A^2 - B^2\)[/tex].
### Conclusion:
The following expressions show a difference of squares:
1. [tex]\(10 y^2 - 4 x^2\)[/tex]
2. [tex]\(16 y^2 - x^2\)[/tex]
In summary, both [tex]\(10 y^2 - 4 x^2\)[/tex] and [tex]\(16 y^2 - x^2\)[/tex] can be written as a difference of squares, while [tex]\(8 x^2 - 40 x + 25\)[/tex] and [tex]\(64 x^2 - 48 x + 9\)[/tex] cannot.
