Simplify (9r)/(r^2-4)-(8r+1)/(r^2-4r+4)

$\frac{9 r}{r^{2} - 4} - \frac{8 r + 1}{r^{2} - 4 r + 4}$

Step 1

Simplify each term.

Simplify the denominator.

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

$\frac{9 r}{r^{2} - 2^{2}} - \frac{8 r + 1}{r^{2} - 4 r + 4}$

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 $b = 2$ .

$\frac{9 r}{(r + 2) (r - 2)} - \frac{8 r + 1}{r^{2} - 4 r + 4}$

Factor using the perfect square rule.

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

$\frac{9 r}{(r + 2) (r - 2)} - \frac{8 r + 1}{r^{2} - 4 r + 2^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 r = 2 \cdot r \cdot 2$

Rewrite the polynomial.

$\frac{9 r}{(r + 2) (r - 2)} - \frac{8 r + 1}{r^{2} - 2 \cdot r \cdot 2 + 2^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = r$ and $b = 2$ .

$\frac{9 r}{(r + 2) (r - 2)} - \frac{8 r + 1}{{(r - 2)}^{2}}$

Step 2

To write $\frac{9 r}{(r + 2) (r - 2)}$ as a fraction with a common denominator, multiply by $\frac{r - 2}{r - 2}$ .

$\frac{9 r}{(r + 2) (r - 2)} \cdot \frac{r - 2}{r - 2} - \frac{8 r + 1}{{(r - 2)}^{2}}$

Step 3

To write $- \frac{8 r + 1}{{(r - 2)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{r + 2}{r + 2}$ .

$\frac{9 r}{(r + 2) (r - 2)} \cdot \frac{r - 2}{r - 2} - \frac{8 r + 1}{{(r - 2)}^{2}} \cdot \frac{r + 2}{r + 2}$

Step 4

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

Multiply $\frac{9 r}{(r + 2) (r - 2)}$ by $\frac{r - 2}{r - 2}$ .

$\frac{9 r (r - 2)}{(r + 2) (r - 2) (r - 2)} - \frac{8 r + 1}{{(r - 2)}^{2}} \cdot \frac{r + 2}{r + 2}$

Raise $r - 2$ to the power of $1$ .

$\frac{9 r (r - 2)}{(r + 2) ({(r - 2)}^{1} (r - 2))} - \frac{8 r + 1}{{(r - 2)}^{2}} \cdot \frac{r + 2}{r + 2}$

Raise $r - 2$ to the power of $1$ .

$\frac{9 r (r - 2)}{(r + 2) ({(r - 2)}^{1} {(r - 2)}^{1})} - \frac{8 r + 1}{{(r - 2)}^{2}} \cdot \frac{r + 2}{r + 2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 r (r - 2)}{(r + 2) {(r - 2)}^{1 + 1}} - \frac{8 r + 1}{{(r - 2)}^{2}} \cdot \frac{r + 2}{r + 2}$

Add $1$ and $1$ .

$\frac{9 r (r - 2)}{(r + 2) {(r - 2)}^{2}} - \frac{8 r + 1}{{(r - 2)}^{2}} \cdot \frac{r + 2}{r + 2}$

Multiply $\frac{8 r + 1}{{(r - 2)}^{2}}$ by $\frac{r + 2}{r + 2}$ .

$\frac{9 r (r - 2)}{(r + 2) {(r - 2)}^{2}} - \frac{(8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Reorder the factors of $(r + 2) {(r - 2)}^{2}$ .

$\frac{9 r (r - 2)}{{(r - 2)}^{2} (r + 2)} - \frac{(8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 r (r - 2) - (8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{9 r \cdot r + 9 r \cdot - 2 - (8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$\frac{9 (r \cdot r) + 9 r \cdot - 2 - (8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Multiply $r$ by $r$ .

$\frac{9 r^{2} + 9 r \cdot - 2 - (8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Multiply $- 2$ by $9$ .

$\frac{9 r^{2} - 18 r - (8 r + 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Apply the distributive property.

$\frac{9 r^{2} - 18 r + (- (8 r) - 1 \cdot 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Multiply $8$ by $- 1$ .

$\frac{9 r^{2} - 18 r + (- 8 r - 1 \cdot 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Multiply $- 1$ by $1$ .

$\frac{9 r^{2} - 18 r + (- 8 r - 1) (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Expand $(- 8 r - 1) (r + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{9 r^{2} - 18 r - 8 r (r + 2) - 1 (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Apply the distributive property.

$\frac{9 r^{2} - 18 r - 8 r \cdot r - 8 r \cdot 2 - 1 (r + 2)}{{(r - 2)}^{2} (r + 2)}$

Apply the distributive property.

$\frac{9 r^{2} - 18 r - 8 r \cdot r - 8 r \cdot 2 - 1 r - 1 \cdot 2}{{(r - 2)}^{2} (r + 2)}$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$\frac{9 r^{2} - 18 r - 8 (r \cdot r) - 8 r \cdot 2 - 1 r - 1 \cdot 2}{{(r - 2)}^{2} (r + 2)}$

Multiply $r$ by $r$ .

$\frac{9 r^{2} - 18 r - 8 r^{2} - 8 r \cdot 2 - 1 r - 1 \cdot 2}{{(r - 2)}^{2} (r + 2)}$

Multiply $2$ by $- 8$ .

$\frac{9 r^{2} - 18 r - 8 r^{2} - 16 r - 1 r - 1 \cdot 2}{{(r - 2)}^{2} (r + 2)}$

Rewrite $- 1 r$ as $- r$ .

$\frac{9 r^{2} - 18 r - 8 r^{2} - 16 r - r - 1 \cdot 2}{{(r - 2)}^{2} (r + 2)}$

Multiply $- 1$ by $2$ .

$\frac{9 r^{2} - 18 r - 8 r^{2} - 16 r - r - 2}{{(r - 2)}^{2} (r + 2)}$

Subtract $r$ from $- 16 r$ .

$\frac{9 r^{2} - 18 r - 8 r^{2} - 17 r - 2}{{(r - 2)}^{2} (r + 2)}$

Subtract $8 r^{2}$ from $9 r^{2}$ .

$\frac{r^{2} - 18 r - 17 r - 2}{{(r - 2)}^{2} (r + 2)}$

Subtract $17 r$ from $- 18 r$ .

$\frac{r^{2} - 35 r - 2}{{(r - 2)}^{2} (r + 2)}$

Simplify (9r)/(r^2-4)-(8r+1)/(r^2-4r+4)

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