Basic Math Examples

$\frac{y^{2} - 4}{y^{2} - 7 y + 12} - \frac{2}{y - 4}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{y^{2} - 2^{2}}{y^{2} - 7 y + 12} - \frac{2}{y - 4}$

Step 1.1.2

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 $b = 2$ .

$\frac{(y + 2) (y - 2)}{y^{2} - 7 y + 12} - \frac{2}{y - 4}$

Step 1.2

Factor $y^{2} - 7 y + 12$ using the AC method.

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Step 1.2.1

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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{(y + 2) (y - 2)}{(y - 4) (y - 3)} - \frac{2}{y - 4}$

Step 2

To write $- \frac{2}{y - 4}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{(y + 2) (y - 2)}{(y - 4) (y - 3)} - \frac{2}{y - 4} \cdot \frac{y - 3}{y - 3}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2}{y - 4}$ by $\frac{y - 3}{y - 3}$ .

$\frac{(y + 2) (y - 2)}{(y - 4) (y - 3)} - \frac{2 (y - 3)}{(y - 4) (y - 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(y + 2) (y - 2) - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(y + 2) (y - 2)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{y (y - 2) + 2 (y - 2) - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.1.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 2 + 2 (y - 2) - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.1.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 2 + 2 y + 2 \cdot - 2 - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.2

Combine the opposite terms in $y \cdot y + y \cdot - 2 + 2 y + 2 \cdot - 2$ .

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Step 4.2.1

Reorder the factors in the terms $y \cdot - 2$ and $2 y$ .

$\frac{y \cdot y - 2 y + 2 y + 2 \cdot - 2 - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.2.2

Add $- 2 y$ and $2 y$ .

$\frac{y \cdot y + 0 + 2 \cdot - 2 - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.2.3

Add $y \cdot y$ and $0$ .

$\frac{y \cdot y + 2 \cdot - 2 - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.3

Simplify each term.

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Step 4.3.1

Multiply $y$ by $y$ .

$\frac{y^{2} + 2 \cdot - 2 - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.3.2

Multiply $2$ by $- 2$ .

$\frac{y^{2} - 4 - 2 (y - 3)}{(y - 4) (y - 3)}$

Step 4.4

Apply the distributive property.

$\frac{y^{2} - 4 - 2 y - 2 \cdot - 3}{(y - 4) (y - 3)}$

Step 4.5

Multiply $- 2$ by $- 3$ .

$\frac{y^{2} - 4 - 2 y + 6}{(y - 4) (y - 3)}$

Step 4.6

Add $- 4$ and $6$ .

$\frac{y^{2} - 2 y + 2}{(y - 4) (y - 3)}$