Simplify (x^2-25y^2)÷(x^2+10xy+25y^2)

$(x^{2} - 25 y^{2}) \div (x^{2} + 10 x y + 25 y^{2})$

Step 1

Rewrite the division as a fraction.

$\frac{x^{2} - 25 y^{2}}{x^{2} + 10 x y + 25 y^{2}}$

Step 2

Simplify the numerator.

Rewrite $25 y^{2}$ as ${(5 y)}^{2}$ .

$\frac{x^{2} - {(5 y)}^{2}}{x^{2} + 10 x y + 25 y^{2}}$

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

$\frac{(x + 5 y) (x - (5 y))}{x^{2} + 10 x y + 25 y^{2}}$

Multiply $5$ by $- 1$ .

$\frac{(x + 5 y) (x - 5 y)}{x^{2} + 10 x y + 25 y^{2}}$

Step 3

Factor using the perfect square rule.

Rewrite $25 y^{2}$ as ${(5 y)}^{2}$ .

$\frac{(x + 5 y) (x - 5 y)}{x^{2} + 10 x y + {(5 y)}^{2}}$

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

$10 x y = 2 \cdot x \cdot (5 y)$

Rewrite the polynomial.

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

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

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

Step 4

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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