Calculus Examples

$- \frac{30 r^{2} - 30}{10 r^{2} + 10}$

Step 1

Cancel the common factor of $30 r^{2} - 30$ and $10 r^{2} + 10$ .

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Step 1.1

Factor $10$ out of $30 r^{2}$ .

$- \frac{10 (3 r^{2}) - 30}{10 r^{2} + 10}$

Step 1.2

Factor $10$ out of $- 30$ .

$- \frac{10 (3 r^{2}) + 10 \cdot - 3}{10 r^{2} + 10}$

Step 1.3

Factor $10$ out of $10 (3 r^{2}) + 10 (- 3)$ .

$- \frac{10 (3 r^{2} - 3)}{10 r^{2} + 10}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $10$ out of $10 r^{2}$ .

$- \frac{10 (3 r^{2} - 3)}{10 (r^{2}) + 10}$

Step 1.4.2

Factor $10$ out of $10$ .

$- \frac{10 (3 r^{2} - 3)}{10 (r^{2}) + 10 (1)}$

Step 1.4.3

Factor $10$ out of $10 (r^{2}) + 10 (1)$ .

$- \frac{10 (3 r^{2} - 3)}{10 (r^{2} + 1)}$

Step 1.4.4

Cancel the common factor.

$- \frac{10 (3 r^{2} - 3)}{10 (r^{2} + 1)}$

Step 1.4.5

Rewrite the expression.

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

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 r^{2} - 3$ .

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Step 2.1.1

Factor $3$ out of $3 r^{2}$ .

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

Step 2.1.2

Factor $3$ out of $- 3$ .

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

Step 2.1.3

Factor $3$ out of $3 (r^{2}) + 3 (- 1)$ .

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

Step 2.2

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

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

Step 2.3

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 $b = 1$ .

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