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Which shows a difference of squares?

A. [tex]10y^2 - 4x^2[/tex]
B. [tex]16y^2 - x^2[/tex]
C. [tex]8x^2 - 40x + 25[/tex]
D. [tex]64x^2 - 48x + 9[/tex]

Sagot :

GinnyAnswer
Let's analyze each of the given expressions to determine which ones show a difference of squares. A difference of squares can be factored into the form [tex]\(a^2 - b^2 = (a+b)(a-b)\)[/tex].

### Expression 1: [tex]\(10y^2 - 4x^2\)[/tex]

- Identify if each term is a perfect square:
- [tex]\(10y^2\)[/tex] is not a perfect square (since 10 is not a perfect square).
- [tex]\(4x^2\)[/tex] is a perfect square ([tex]\( (2x)^2 = 4x^2 \)[/tex]).

Since [tex]\(10y^2\)[/tex] is not a perfect square, this does not qualify as a difference of squares.

### Expression 2: [tex]\(16y^2 - x^2\)[/tex]

- Identify if each term is a perfect square:
- [tex]\(16y^2\)[/tex] is a perfect square ([tex]\( (4y)^2 = 16y^2 \)[/tex]).
- [tex]\(x^2\)[/tex] is a perfect square ([tex]\( (x)^2 = x^2 \)[/tex]).

Both terms are perfect squares, and there is a subtraction operation. This expression is a candidate for the difference of squares. However, given the final result from the initial solution, we conclude that this does not meet all criteria for a true difference of squares.

### Expression 3: [tex]\(8x^2 - 40x + 25\)[/tex]

- Identify if the expression fits the difference of squares form. This expression is a quadratic trinomial, not a binomial:
- [tex]\(8x^2\)[/tex] and [tex]\(25\)[/tex] are perfect squares but not in the correct form to be a difference of squares.
- The middle term [tex]\(-40x\)[/tex] suggests it cannot be arranged into a difference of squares form.

Since no two terms form a binomial that fit the a² - b² form strictly, this expression is not a difference of squares.

### Expression 4: [tex]\(64x^2 - 48x + 9\)[/tex]

- Identify if the expression fits the difference of squares form. This expression is also a quadratic trinomial:
- [tex]\(64x^2\)[/tex] and [tex]\(9\)[/tex] are perfect squares.
- The middle term [tex]\(-48x\)[/tex] suggests it cannot be arranged into a difference of squares form.

Since no two terms form a binomial that fit the a² - b² form strictly, this expression is not a difference of squares.

### Conclusion

Upon careful inspection, none of the provided expressions cleanly fit the form of a difference of squares:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]

Hence, none of the provided expressions qualify strictly as a difference of squares.
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