Basic Math Examples

$\frac{3 y^{2} - 3}{y^{2} + 8 y + 7}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3$ out of $3 y^{2} - 3$ .

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Step 1.1.1

Factor $3$ out of $3 y^{2}$ .

$\frac{3 (y^{2}) - 3}{y^{2} + 8 y + 7}$

Step 1.1.2

Factor $3$ out of $- 3$ .

$\frac{3 (y^{2}) + 3 (- 1)}{y^{2} + 8 y + 7}$

Step 1.1.3

Factor $3$ out of $3 (y^{2}) + 3 (- 1)$ .

$\frac{3 (y^{2} - 1)}{y^{2} + 8 y + 7}$

Step 1.2

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

$\frac{3 (y^{2} - 1^{2})}{y^{2} + 8 y + 7}$

Step 1.3

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 $b = 1$ .

$\frac{3 (y + 1) (y - 1)}{y^{2} + 8 y + 7}$

Step 2

Factor $y^{2} + 8 y + 7$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $8$ .

$1, 7$

Step 2.2

Write the factored form using these integers.

$\frac{3 (y + 1) (y - 1)}{(y + 1) (y + 7)}$

Step 3

Cancel the common factor of $y + 1$ .

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Step 3.1

Cancel the common factor.

$\frac{3 (y + 1) (y - 1)}{(y + 1) (y + 7)}$

Step 3.2

Rewrite the expression.

$\frac{3 (y - 1)}{y + 7}$