Basic Math Examples

$\frac{9 r^{2} - 1}{r^{2} - 4} \cdot \frac{2 - r}{3 r + 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $9 r^{2}$ as ${(3 r)}^{2}$ .

$\frac{{(3 r)}^{2} - 1}{r^{2} - 4} \cdot \frac{2 - r}{3 r + 1}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(3 r)}^{2} - 1^{2}}{r^{2} - 4} \cdot \frac{2 - r}{3 r + 1}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 r$ and $b = 1$ .

$\frac{(3 r + 1) (3 r - 1)}{r^{2} - 4} \cdot \frac{2 - r}{3 r + 1}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{(3 r + 1) (3 r - 1)}{r^{2} - 2^{2}} \cdot \frac{2 - r}{3 r + 1}$

Step 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 $b = 2$ .

$\frac{(3 r + 1) (3 r - 1)}{(r + 2) (r - 2)} \cdot \frac{2 - r}{3 r + 1}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $3 r + 1$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(3 r + 1) (3 r - 1)}{(r + 2) (r - 2)} \cdot \frac{2 - r}{3 r + 1}$

Step 3.1.2

Rewrite the expression.

$\frac{3 r - 1}{(r + 2) (r - 2)} \cdot (2 - r)$

Step 3.2

Multiply $\frac{3 r - 1}{(r + 2) (r - 2)}$ by $2 - r$ .

$\frac{(3 r - 1) (2 - r)}{(r + 2) (r - 2)}$

Step 3.3

Cancel the common factor of $2 - r$ and $r - 2$ .

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Step 3.3.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(3 r - 1) (- 1 (- 2) - r)}{(r + 2) (r - 2)}$

Step 3.3.2

Factor $- 1$ out of $- r$ .

$\frac{(3 r - 1) (- 1 (- 2) - (r))}{(r + 2) (r - 2)}$

Step 3.3.3

Factor $- 1$ out of $- 1 (- 2) - (r)$ .

$\frac{(3 r - 1) (- 1 (- 2 + r))}{(r + 2) (r - 2)}$

Step 3.3.4

Reorder terms.

$\frac{(3 r - 1) (- 1 (r - 2))}{(r + 2) (r - 2)}$

Step 3.3.5

Cancel the common factor.

$\frac{(3 r - 1) (- 1 (r - 2))}{(r + 2) (r - 2)}$

Step 3.3.6

Rewrite the expression.

$\frac{(3 r - 1) \cdot (- 1)}{r + 2}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Move $- 1$ to the left of $3 r - 1$ .

$\frac{- 1 \cdot (3 r - 1)}{r + 2}$

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{3 r - 1}{r + 2}$