Basic Math Examples

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

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

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

$r$ .

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