Basic Math Examples

$\frac{5 r}{r^{2} + 2 r - 35} - \frac{r}{r^{2} - 49}$

Step 1

Simplify each term.

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Step 1.1

Factor $r^{2} + 2 r - 35$ using the AC method.

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Step 1.1.1

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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 1.1.2

Write the factored form using these integers.

$\frac{5 r}{(r - 5) (r + 7)} - \frac{r}{r^{2} - 49}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $49$ as $7^{2}$ .

$\frac{5 r}{(r - 5) (r + 7)} - \frac{r}{r^{2} - 7^{2}}$

Step 1.2.2

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 $b = 7$ .

$\frac{5 r}{(r - 5) (r + 7)} - \frac{r}{(r + 7) (r - 7)}$

Step 2

To write $\frac{5 r}{(r - 5) (r + 7)}$ as a fraction with a common denominator, multiply by $\frac{r - 7}{r - 7}$ .

$\frac{5 r}{(r - 5) (r + 7)} \cdot \frac{r - 7}{r - 7} - \frac{r}{(r + 7) (r - 7)}$

Step 3

To write $- \frac{r}{(r + 7) (r - 7)}$ as a fraction with a common denominator, multiply by $\frac{r - 5}{r - 5}$ .

$\frac{5 r}{(r - 5) (r + 7)} \cdot \frac{r - 7}{r - 7} - \frac{r}{(r + 7) (r - 7)} \cdot \frac{r - 5}{r - 5}$

Step 4

Write each expression with a common denominator of $(r - 5) (r + 7) (r - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{5 r}{(r - 5) (r + 7)}$ by $\frac{r - 7}{r - 7}$ .

$\frac{5 r (r - 7)}{(r - 5) (r + 7) (r - 7)} - \frac{r}{(r + 7) (r - 7)} \cdot \frac{r - 5}{r - 5}$

Step 4.2

Multiply $\frac{r}{(r + 7) (r - 7)}$ by $\frac{r - 5}{r - 5}$ .

$\frac{5 r (r - 7)}{(r - 5) (r + 7) (r - 7)} - \frac{r (r - 5)}{(r + 7) (r - 7) (r - 5)}$

Step 4.3

Reorder the factors of $(r - 5) (r + 7) (r - 7)$ .

$\frac{5 r (r - 7)}{(r + 7) (r - 5) (r - 7)} - \frac{r (r - 5)}{(r + 7) (r - 7) (r - 5)}$

Step 4.4

Reorder the factors of $(r + 7) (r - 7) (r - 5)$ .

$\frac{5 r (r - 7)}{(r + 7) (r - 5) (r - 7)} - \frac{r (r - 5)}{(r + 7) (r - 5) (r - 7)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 r (r - 7) - r (r - 5)}{(r + 7) (r - 5) (r - 7)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $r$ out of $5 r (r - 7) - r (r - 5)$ .

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Step 6.1.1

Factor $r$ out of $5 r (r - 7)$ .

$\frac{r (5 (r - 7)) - r (r - 5)}{(r + 7) (r - 5) (r - 7)}$

Step 6.1.2

Factor $r$ out of $- r (r - 5)$ .

$\frac{r (5 (r - 7)) + r (- (r - 5))}{(r + 7) (r - 5) (r - 7)}$

Step 6.1.3

Factor $r$ out of $r (5 (r - 7)) + r (- (r - 5))$ .

$\frac{r (5 (r - 7) - (r - 5))}{(r + 7) (r - 5) (r - 7)}$

Step 6.2

Apply the distributive property.

$\frac{r (5 r + 5 \cdot - 7 - (r - 5))}{(r + 7) (r - 5) (r - 7)}$

Step 6.3

Multiply $5$ by $- 7$ .

$\frac{r (5 r - 35 - (r - 5))}{(r + 7) (r - 5) (r - 7)}$

Step 6.4

Apply the distributive property.

$\frac{r (5 r - 35 - r - - 5)}{(r + 7) (r - 5) (r - 7)}$

Step 6.5

Multiply $- 1$ by $- 5$ .

$\frac{r (5 r - 35 - r + 5)}{(r + 7) (r - 5) (r - 7)}$

Step 6.6

Subtract $r$ from $5 r$ .

$\frac{r (4 r - 35 + 5)}{(r + 7) (r - 5) (r - 7)}$

Step 6.7

Add $- 35$ and $5$ .

$\frac{r (4 r - 30)}{(r + 7) (r - 5) (r - 7)}$

Step 6.8

Factor $2$ out of $4 r - 30$ .

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Step 6.8.1

Factor $2$ out of $4 r$ .

$\frac{r (2 (2 r) - 30)}{(r + 7) (r - 5) (r - 7)}$

Step 6.8.2

Factor $2$ out of $- 30$ .

$\frac{r (2 (2 r) + 2 (- 15))}{(r + 7) (r - 5) (r - 7)}$

Step 6.8.3

Factor $2$ out of $2 (2 r) + 2 (- 15)$ .

$\frac{r (2 (2 r - 15))}{(r + 7) (r - 5) (r - 7)}$

$\frac{r \cdot 2 (2 r - 15)}{(r + 7) (r - 5) (r - 7)}$

Step 7

Move $2$ to the left of $r$ .

$\frac{2 r (2 r - 15)}{(r + 7) (r - 5) (r - 7)}$