To determine which expressions are differences of squares, let's first recall what a difference of squares is. An expression [tex]\(A^2 - B^2\)[/tex] is called a difference of squares because it can be factored into [tex]\((A - B)(A + B)\)[/tex].
Given the expressions:
1. [tex]\(t^2 - 4\)[/tex]
2. [tex]\(h^2 - 20\)[/tex]
3. [tex]\(a^2 + 1\)[/tex]
4. [tex]\(x^2 - 169\)[/tex]
We need to check each expression to see if it fits the form [tex]\(A^2 - B^2\)[/tex].
1. Expression: [tex]\(t^2 - 4\)[/tex]
- [tex]\(t^2 - 4\)[/tex] can be written as [tex]\(t^2 - 2^2\)[/tex].
- This fits the form [tex]\(A^2 - B^2\)[/tex] with [tex]\(A = t\)[/tex] and [tex]\(B = 2\)[/tex].
- Hence, [tex]\(t^2 - 4\)[/tex] is a difference of squares.
2. Expression: [tex]\(h^2 - 20\)[/tex]
- [tex]\(h^2 - 20\)[/tex] is not immediately obvious as a difference of squares.
- [tex]\(20\)[/tex] is not a perfect square, so it does not fit the form [tex]\(A^2 - B^2\)[/tex].
- Hence, [tex]\(h^2 - 20\)[/tex] is not a difference of squares.
3. Expression: [tex]\(a^2 + 1\)[/tex]
- [tex]\(a^2 + 1\)[/tex] is a sum, not a difference.
- It does not fit the form [tex]\(A^2 - B^2\)[/tex].
- Hence, [tex]\(a^2 + 1\)[/tex] is not a difference of squares.
4. Expression: [tex]\(x^2 - 169\)[/tex]
- [tex]\(x^2 - 169\)[/tex] can be written as [tex]\(x^2 - 13^2\)[/tex].
- This fits the form [tex]\(A^2 - B^2\)[/tex] with [tex]\(A = x\)[/tex] and [tex]\(B = 13\)[/tex].
- Hence, [tex]\(x^2 - 169\)[/tex] is a difference of squares.
After analyzing all the expressions, the one that does not fit the difference of squares pattern is:
- [tex]\(h^2 - 20\)[/tex]
Therefore, the expressions that are differences of squares are:
- [tex]\(t^2 - 4\)[/tex]
- [tex]\(x^2 - 169\)[/tex]
Following the precise steps, it can be concluded that none of the given expressions, based on the provided work, was considered as a difference of squares.
So, the final expression identified as a difference of squares is:
- [tex]\(h^2 - 20\)[/tex]
