Answer :
To determine which of the given expressions qualify as a difference of squares, we need to recognize the form of a difference of squares:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Our goal is to see if we can rewrite any of the given expressions into this form. Let's analyze each of the expressions one by one:
1. Expression: [tex]\( 10y^2 - 4x^2 \)[/tex]
To see if this fits the difference of squares form, we rewrite it as:
[tex]\[ 10y^2 - 4x^2 = ( \sqrt{10}y)^2 - (2x)^2 \][/tex]
Since we now have the form [tex]\( a^2 - b^2 \)[/tex]:
[tex]\[ a = \sqrt{10}y \quad \text{and} \quad b = 2x \][/tex]
This expression [tex]\( 10y^2 - 4x^2 \)[/tex] is indeed a difference of squares.
2. Expression: [tex]\( 16y^2 - x^2 \)[/tex]
Rewriting this as:
[tex]\[ 16y^2 - x^2 = (4y)^2 - (x)^2 \][/tex]
This fits the form [tex]\( a^2 - b^2 \)[/tex]:
[tex]\[ a = 4y \quad \text{and} \quad b = x \][/tex]
Thus, [tex]\( 16y^2 - x^2 \)[/tex] is a difference of squares.
3. Expression: [tex]\( 8x^2 - 40x + 25 \)[/tex]
We check if this fits the form [tex]\( a^2 - b^2 \)[/tex]. First, we notice the presence of a linear term which suggests it may not be a difference of squares. However, it can be factored differently:
[tex]\[ 8x^2 - 40x + 25 = (2x - 5)^2 \][/tex]
This is actually a perfect square trinomial, not a difference of squares. So, [tex]\( 8x^2 - 40x + 25 \)[/tex] is not a difference of squares.
4. Expression: [tex]\( 64x^2 - 48x + 9 \)[/tex]
Similar to the previous expression, the linear middle term suggests it’s not a difference of squares but a perfect square trinomial:
[tex]\[ 64x^2 - 48x + 9 = (8x - 3)^2 \][/tex]
This is another example of a perfect square trinomial, not a difference of squares. So, [tex]\( 64x^2 - 48x + 9 \)[/tex] is not a difference of squares.
Conclusion:
From the analysis, the expressions showing a difference of squares are:
[tex]\[ 10y^2 - 4x^2 \][/tex]
[tex]\[ 16y^2 - x^2 \][/tex]
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Our goal is to see if we can rewrite any of the given expressions into this form. Let's analyze each of the expressions one by one:
1. Expression: [tex]\( 10y^2 - 4x^2 \)[/tex]
To see if this fits the difference of squares form, we rewrite it as:
[tex]\[ 10y^2 - 4x^2 = ( \sqrt{10}y)^2 - (2x)^2 \][/tex]
Since we now have the form [tex]\( a^2 - b^2 \)[/tex]:
[tex]\[ a = \sqrt{10}y \quad \text{and} \quad b = 2x \][/tex]
This expression [tex]\( 10y^2 - 4x^2 \)[/tex] is indeed a difference of squares.
2. Expression: [tex]\( 16y^2 - x^2 \)[/tex]
Rewriting this as:
[tex]\[ 16y^2 - x^2 = (4y)^2 - (x)^2 \][/tex]
This fits the form [tex]\( a^2 - b^2 \)[/tex]:
[tex]\[ a = 4y \quad \text{and} \quad b = x \][/tex]
Thus, [tex]\( 16y^2 - x^2 \)[/tex] is a difference of squares.
3. Expression: [tex]\( 8x^2 - 40x + 25 \)[/tex]
We check if this fits the form [tex]\( a^2 - b^2 \)[/tex]. First, we notice the presence of a linear term which suggests it may not be a difference of squares. However, it can be factored differently:
[tex]\[ 8x^2 - 40x + 25 = (2x - 5)^2 \][/tex]
This is actually a perfect square trinomial, not a difference of squares. So, [tex]\( 8x^2 - 40x + 25 \)[/tex] is not a difference of squares.
4. Expression: [tex]\( 64x^2 - 48x + 9 \)[/tex]
Similar to the previous expression, the linear middle term suggests it’s not a difference of squares but a perfect square trinomial:
[tex]\[ 64x^2 - 48x + 9 = (8x - 3)^2 \][/tex]
This is another example of a perfect square trinomial, not a difference of squares. So, [tex]\( 64x^2 - 48x + 9 \)[/tex] is not a difference of squares.
Conclusion:
From the analysis, the expressions showing a difference of squares are:
[tex]\[ 10y^2 - 4x^2 \][/tex]
[tex]\[ 16y^2 - x^2 \][/tex]
