Answer :
Let's factor the expression [tex]\( x^2 - 26x + 169 - y^2 \)[/tex].
1. Identify the Perfect Square Trinomial:
First, focus on the expression [tex]\( x^2 - 26x + 169 \)[/tex]. We can see if this part is a perfect square trinomial.
The expression [tex]\( x^2 - 26x + 169 \)[/tex] can be rewritten as:
[tex]\[
(x - 13)^2
\][/tex]
because when expanded, this gives us:
[tex]\[
(x - 13)(x - 13) = x^2 - 13x - 13x + 169 = x^2 - 26x + 169
\][/tex]
2. Write the Entire Expression as a Difference of Squares:
The expression [tex]\( x^2 - 26x + 169 - y^2 \)[/tex] now becomes:
[tex]\[
(x - 13)^2 - y^2
\][/tex]
3. Apply the Difference of Squares Formula:
A difference of squares is expressed as [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
Here, [tex]\( a = x - 13 \)[/tex] and [tex]\( b = y \)[/tex]. So we have:
[tex]\[
(x - 13 - y)(x - 13 + y)
\][/tex]
Therefore, the factored form of the expression [tex]\( x^2 - 26x + 169 - y^2 \)[/tex] is:
[tex]\[
(x - 13 - y)(x - 13 + y)
\][/tex]
This matches the solution choice provided.
1. Identify the Perfect Square Trinomial:
First, focus on the expression [tex]\( x^2 - 26x + 169 \)[/tex]. We can see if this part is a perfect square trinomial.
The expression [tex]\( x^2 - 26x + 169 \)[/tex] can be rewritten as:
[tex]\[
(x - 13)^2
\][/tex]
because when expanded, this gives us:
[tex]\[
(x - 13)(x - 13) = x^2 - 13x - 13x + 169 = x^2 - 26x + 169
\][/tex]
2. Write the Entire Expression as a Difference of Squares:
The expression [tex]\( x^2 - 26x + 169 - y^2 \)[/tex] now becomes:
[tex]\[
(x - 13)^2 - y^2
\][/tex]
3. Apply the Difference of Squares Formula:
A difference of squares is expressed as [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
Here, [tex]\( a = x - 13 \)[/tex] and [tex]\( b = y \)[/tex]. So we have:
[tex]\[
(x - 13 - y)(x - 13 + y)
\][/tex]
Therefore, the factored form of the expression [tex]\( x^2 - 26x + 169 - y^2 \)[/tex] is:
[tex]\[
(x - 13 - y)(x - 13 + y)
\][/tex]
This matches the solution choice provided.
