Trigonometry Examples

$\frac{4 x^{2} - 13 x + 10}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 4 \cdot 10 = 40$ and whose sum is $b = - 13$ .

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Step 1.1.1

Factor $- 13$ out of $- 13 x$ .

$\frac{4 x^{2} - 13 (x) + 10}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 1.1.2

Rewrite $- 13$ as $- 5$ plus $- 8$

$\frac{4 x^{2} + (- 5 - 8) x + 10}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 1.1.3

Apply the distributive property.

$\frac{4 x^{2} - 5 x - 8 x + 10}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(4 x^{2} - 5 x) - 8 x + 10}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 1.2.2

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

$\frac{x (4 x - 5) - 2 (4 x - 5)}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 5$ .

$\frac{(4 x - 5) (x - 2)}{24 x^{2} - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $6 x$ out of $24 x^{2} - 30 x$ .

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Step 2.1.1

Factor $6 x$ out of $24 x^{2}$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x) - 30 x} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 2.1.2

Factor $6 x$ out of $- 30 x$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x) + 6 x (- 5)} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 2.1.3

Factor $6 x$ out of $6 x (4 x) + 6 x (- 5)$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x^{2} + 8 x}{2 x^{2} - 8}$

Step 2.2

Factor $4 x$ out of $4 x^{2} + 8 x$ .

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Step 2.2.1

Factor $4 x$ out of $4 x^{2}$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x) + 8 x}{2 x^{2} - 8}$

Step 2.2.2

Factor $4 x$ out of $8 x$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x) + 4 x (2)}{2 x^{2} - 8}$

Step 2.2.3

Factor $4 x$ out of $4 x (x) + 4 x (2)$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 x^{2} - 8}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $2$ out of $2 x^{2} - 8$ .

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Step 3.1.1

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

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 (x^{2}) - 8}$

Step 3.1.2

Factor $2$ out of $- 8$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 x^{2} + 2 \cdot - 4}$

Step 3.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 (x^{2} - 4)}$

Step 3.2

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

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 (x^{2} - 2^{2})}$

Step 3.3

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

$\frac{(4 x - 5) (x - 2)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 (x + 2) (x - 2)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 2$ .

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Step 4.1.1

Factor $x - 2$ out of $(4 x - 5) (x - 2)$ .

$\frac{(x - 2) (4 x - 5)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 (x + 2) (x - 2)}$

Step 4.1.2

Factor $x - 2$ out of $2 (x + 2) (x - 2)$ .

$\frac{(x - 2) (4 x - 5)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{(x - 2) (2 (x + 2))}$

Step 4.1.3

Cancel the common factor.

$\frac{(x - 2) (4 x - 5)}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{(x - 2) (2 (x + 2))}$

Step 4.1.4

Rewrite the expression.

$\frac{4 x - 5}{6 x (4 x - 5)} \cdot \frac{4 x (x + 2)}{2 (x + 2)}$

Step 4.2

Cancel the common factor of $2 x$ .

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Step 4.2.1

Factor $2 x$ out of $6 x (4 x - 5)$ .

$\frac{4 x - 5}{2 x (3 (4 x - 5))} \cdot \frac{4 x (x + 2)}{2 (x + 2)}$

Step 4.2.2

Factor $2 x$ out of $4 x (x + 2)$ .

$\frac{4 x - 5}{2 x (3 (4 x - 5))} \cdot \frac{2 x (2 (x + 2))}{2 (x + 2)}$

Step 4.2.3

Cancel the common factor.

$\frac{4 x - 5}{2 x (3 (4 x - 5))} \cdot \frac{2 x (2 (x + 2))}{2 (x + 2)}$

Step 4.2.4

Rewrite the expression.

$\frac{4 x - 5}{3 (4 x - 5)} \cdot \frac{2 (x + 2)}{2 (x + 2)}$

Step 4.3

Multiply $\frac{4 x - 5}{3 (4 x - 5)}$ by $\frac{2 (x + 2)}{2 (x + 2)}$ .

$\frac{(4 x - 5) (2 (x + 2))}{3 (4 x - 5) (2 (x + 2))}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{(4 x - 5) (2 (x + 2))}{6 (4 x - 5) (x + 2)}$

Step 4.5

Cancel the common factor of $4 x - 5$ .

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Step 4.5.1

Cancel the common factor.

$\frac{(4 x - 5) (2 (x + 2))}{6 (4 x - 5) (x + 2)}$

Step 4.5.2

Rewrite the expression.

$\frac{2 (x + 2)}{6 (x + 2)}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $2$ out of $6 (x + 2)$ .

$\frac{2 (x + 2)}{2 (3 (x + 2))}$

Step 5.2

Cancel the common factor.

$\frac{2 (x + 2)}{2 (3 (x + 2))}$

Step 5.3

Rewrite the expression.

$\frac{x + 2}{3 (x + 2)}$

Step 6

Cancel the common factor of $x + 2$ .

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Step 6.1

Cancel the common factor.

$\frac{x + 2}{3 (x + 2)}$

Step 6.2

Rewrite the expression.

$\frac{1}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$