Trigonometry Examples

$\frac{9 - x^{2}}{10 x^{2} - 28 x - 6}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - x^{2}}{10 x^{2} - 28 x - 6}$

Step 1.2

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

$\frac{(3 + x) (3 - x)}{10 x^{2} - 28 x - 6}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $10 x^{2} - 28 x - 6$ .

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Step 2.1.1

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

$\frac{(3 + x) (3 - x)}{2 (5 x^{2}) - 28 x - 6}$

Step 2.1.2

Factor $2$ out of $- 28 x$ .

$\frac{(3 + x) (3 - x)}{2 (5 x^{2}) + 2 (- 14 x) - 6}$

Step 2.1.3

Factor $2$ out of $- 6$ .

$\frac{(3 + x) (3 - x)}{2 (5 x^{2}) + 2 (- 14 x) + 2 (- 3)}$

Step 2.1.4

Factor $2$ out of $2 (5 x^{2}) + 2 (- 14 x)$ .

$\frac{(3 + x) (3 - x)}{2 (5 x^{2} - 14 x) + 2 (- 3)}$

Step 2.1.5

Factor $2$ out of $2 (5 x^{2} - 14 x) + 2 (- 3)$ .

$\frac{(3 + x) (3 - x)}{2 (5 x^{2} - 14 x - 3)}$

Step 2.2

Factor by grouping.

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Step 2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 3 = - 15$ and whose sum is $b = - 14$ .

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Step 2.2.1.1

Factor $- 14$ out of $- 14 x$ .

$\frac{(3 + x) (3 - x)}{2 (5 x^{2} - 14 (x) - 3)}$

Step 2.2.1.2

Rewrite $- 14$ as $1$ plus $- 15$

$\frac{(3 + x) (3 - x)}{2 (5 x^{2} + (1 - 15) x - 3)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{(3 + x) (3 - x)}{2 (5 x^{2} + 1 x - 15 x - 3)}$

Step 2.2.1.4

Multiply $x$ by $1$ .

$\frac{(3 + x) (3 - x)}{2 (5 x^{2} + x - 15 x - 3)}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{(3 + x) (3 - x)}{2 ((5 x^{2} + x) - 15 x - 3)}$

Step 2.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(3 + x) (3 - x)}{2 (x (5 x + 1) - 3 (5 x + 1))}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

$\frac{(3 + x) (3 - x)}{2 ((5 x + 1) (x - 3))}$

$\frac{(3 + x) (3 - x)}{2 (5 x + 1) (x - 3)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 3.1.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(3 + x) (- 1 (- 3) - x)}{2 (5 x + 1) (x - 3)}$

Step 3.1.2

Factor $- 1$ out of $- x$ .

$\frac{(3 + x) (- 1 (- 3) - (x))}{2 (5 x + 1) (x - 3)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{(3 + x) (- 1 (- 3 + x))}{2 (5 x + 1) (x - 3)}$

Step 3.1.4

Reorder terms.

$\frac{(3 + x) (- 1 (x - 3))}{2 (5 x + 1) (x - 3)}$

Step 3.1.5

Cancel the common factor.

$\frac{(3 + x) (- 1 (x - 3))}{2 (5 x + 1) (x - 3)}$

Step 3.1.6

Rewrite the expression.

$\frac{(3 + x) \cdot (- 1)}{2 (5 x + 1)}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $3 + x$ .

$\frac{- 1 \cdot (3 + x)}{2 (5 x + 1)}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{3 + x}{2 (5 x + 1)}$