Algebra Examples

$\frac{r}{r^{2} - 4} + \frac{1}{r^{2} - 6 r + 8}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{r}{r^{2} - 2^{2}} + \frac{1}{r^{2} - 6 r + 8}$

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 $a = r$ and $b = 2$ .

$\frac{r}{(r + 2) (r - 2)} + \frac{1}{r^{2} - 6 r + 8}$

Step 1.2

Factor $r^{2} - 6 r + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{r}{(r + 2) (r - 2)} + \frac{1}{(r - 4) (r - 2)}$

Step 2

To write $\frac{r}{(r + 2) (r - 2)}$ as a fraction with a common denominator, multiply by $\frac{r - 4}{r - 4}$ .

$\frac{r}{(r + 2) (r - 2)} \cdot \frac{r - 4}{r - 4} + \frac{1}{(r - 4) (r - 2)}$

Step 3

To write $\frac{1}{(r - 4) (r - 2)}$ as a fraction with a common denominator, multiply by $\frac{r + 2}{r + 2}$ .

$\frac{r}{(r + 2) (r - 2)} \cdot \frac{r - 4}{r - 4} + \frac{1}{(r - 4) (r - 2)} \cdot \frac{r + 2}{r + 2}$

Step 4

Write each expression with a common denominator of $(r + 2) (r - 2) (r - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{r}{(r + 2) (r - 2)}$ by $\frac{r - 4}{r - 4}$ .

$\frac{r (r - 4)}{(r + 2) (r - 2) (r - 4)} + \frac{1}{(r - 4) (r - 2)} \cdot \frac{r + 2}{r + 2}$

Step 4.2

Multiply $\frac{1}{(r - 4) (r - 2)}$ by $\frac{r + 2}{r + 2}$ .

$\frac{r (r - 4)}{(r + 2) (r - 2) (r - 4)} + \frac{r + 2}{(r - 4) (r - 2) (r + 2)}$

Step 4.3

Reorder the factors of $(r - 4) (r - 2) (r + 2)$ .

$\frac{r (r - 4)}{(r + 2) (r - 2) (r - 4)} + \frac{r + 2}{(r + 2) (r - 2) (r - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{r (r - 4) + r + 2}{(r + 2) (r - 2) (r - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $r$ by $r$ .

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

Step 6.3

Move $- 4$ to the left of $r$ .

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

Step 6.4

Add $- 4 r$ and $r$ .

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

Step 6.5

Factor $r^{2} - 3 r + 2$ using the AC method.

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Step 6.5.1

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 6.5.2

Write the factored form using these integers.

$\frac{(r - 2) (r - 1)}{(r + 2) (r - 2) (r - 4)}$

Step 7

Cancel the common factor of $r - 2$ .

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Step 7.1

Cancel the common factor.

$\frac{(r - 2) (r - 1)}{(r + 2) (r - 2) (r - 4)}$

Step 7.2

Rewrite the expression.

$\frac{r - 1}{(r + 2) (r - 4)}$