Basic Math Examples

$\frac{y^{2}}{y^{2} - 1} + \frac{2 y + 1}{y^{2} - 1}$

Step 1

Combine the numerators over the common denominator.

$\frac{y^{2} + 2 y + 1}{y^{2} - 1}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{y^{2} + 2 y + 1^{2}}{y^{2} - 1}$

Step 2.2

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

$2 y = 2 \cdot y \cdot 1$

Step 2.3

Rewrite the polynomial.

$\frac{y^{2} + 2 \cdot y \cdot 1 + 1^{2}}{y^{2} - 1}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 1$ .

$\frac{{(y + 1)}^{2}}{y^{2} - 1}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\frac{{(y + 1)}^{2}}{y^{2} - 1^{2}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{{(y + 1)}^{2}}{(y + 1) (y - 1)}$

Step 4

Cancel the common factor of ${(y + 1)}^{2}$ and $y + 1$ .

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Step 4.1

Factor $y + 1$ out of ${(y + 1)}^{2}$ .

$\frac{(y + 1) (y + 1)}{(y + 1) (y - 1)}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Cancel the common factor.

$\frac{(y + 1) (y + 1)}{(y + 1) (y - 1)}$

Step 4.2.2

Rewrite the expression.

$\frac{y + 1}{y - 1}$