Simplify (4y^2-25)/(4y^2-20y+25)

$\frac{4 y^{2} - 25}{4 y^{2} - 20 y + 25}$

Simplify the numerator.

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{{(2 y)}^{2} - 25}{4 y^{2} - 20 y + 25}$

Rewrite $25$ as $5^{2}$ .

$\frac{{(2 y)}^{2} - 5^{2}}{4 y^{2} - 20 y + 25}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 y$ and $b = 5$ .

$\frac{(2 y + 5) (2 y - 5)}{4 y^{2} - 20 y + 25}$

Factor using the perfect square rule.

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{(2 y + 5) (2 y - 5)}{{(2 y)}^{2} - 20 y + 25}$

Rewrite $25$ as $5^{2}$ .

$\frac{(2 y + 5) (2 y - 5)}{{(2 y)}^{2} - 20 y + 5^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$20 y = 2 \cdot (2 y) \cdot 5$

Rewrite the polynomial.

$\frac{(2 y + 5) (2 y - 5)}{{(2 y)}^{2} - 2 \cdot (2 y) \cdot 5 + 5^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 y$ and $b = 5$ .

$\frac{(2 y + 5) (2 y - 5)}{{(2 y - 5)}^{2}}$

Cancel the common factor of $2 y - 5$ and ${(2 y - 5)}^{2}$ .

Factor $2 y - 5$ out of $(2 y + 5) (2 y - 5)$ .

$\frac{(2 y - 5) (2 y + 5)}{{(2 y - 5)}^{2}}$

Cancel the common factors.

Factor $2 y - 5$ out of ${(2 y - 5)}^{2}$ .

$\frac{(2 y - 5) (2 y + 5)}{(2 y - 5) (2 y - 5)}$

Cancel the common factor.

$\frac{(2 y - 5) (2 y + 5)}{(2 y - 5) (2 y - 5)}$

Rewrite the expression.

$\frac{2 y + 5}{2 y - 5}$