To simplify the expression
[tex]\[
\frac{y^2 - 10y + 25}{4y^2 - 100}
\][/tex]
we need to factor both the numerator and the denominator and then simplify the fraction by dividing out any common factors.
1. Factor the numerator:
The numerator is [tex]\(y^2 - 10y + 25\)[/tex]. This is a quadratic expression which can be factored as:
[tex]\[
y^2 - 10y + 25 = (y - 5)(y - 5) = (y - 5)^2
\][/tex]
2. Factor the denominator:
The denominator is [tex]\(4y^2 - 100\)[/tex]. This is a difference of squares, which can be factored as follows:
[tex]\[
4y^2 - 100 = (2y)^2 - 10^2 = (2y - 10)(2y + 10)
\][/tex]
We can further simplify [tex]\(2y - 10\)[/tex] as:
[tex]\[
2y - 10 = 2(y - 5)
\][/tex]
and similarly, we can express [tex]\(2y + 10\)[/tex]:
[tex]\[
2y + 10 = 2(y + 5)
\][/tex]
Thus, the denominator becomes:
[tex]\[
4y^2 - 100 = 2(y - 5) \cdot 2(y + 5) = 4(y - 5)(y + 5)
\][/tex]
3. Write the fraction with the factored forms:
Now, we can express the original fraction in terms of its factors:
[tex]\[
\frac{(y - 5)^2}{4(y - 5)(y + 5)}
\][/tex]
4. Simplify the fraction:
We see that [tex]\((y - 5)\)[/tex] is a common factor in the numerator and the denominator. We can cancel out one [tex]\((y - 5)\)[/tex] from the numerator and one [tex]\((y - 5)\)[/tex] from the denominator.
[tex]\[
\frac{(y - 5)^2}{4(y - 5)(y + 5)} = \frac{y - 5}{4(y + 5)}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\frac{y - 5}{4(y + 5)}
\][/tex]
