Trigonometry Examples

$\frac{5 p^{2} + 20 p + 15}{25 p^{2} - 50 p - 75}$

Step 1

Cancel the common factor of $5 p^{2} + 20 p + 15$ and $25 p^{2} - 50 p - 75$ .

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Step 1.1

Factor $5$ out of $5 p^{2}$ .

$\frac{5 (p^{2}) + 20 p + 15}{25 p^{2} - 50 p - 75}$

Step 1.2

Factor $5$ out of $20 p$ .

$\frac{5 (p^{2}) + 5 (4 p) + 15}{25 p^{2} - 50 p - 75}$

Step 1.3

Factor $5$ out of $5 (p^{2}) + 5 (4 p)$ .

$\frac{5 (p^{2} + 4 p) + 15}{25 p^{2} - 50 p - 75}$

Step 1.4

Factor $5$ out of $15$ .

$\frac{5 (p^{2} + 4 p) + 5 \cdot 3}{25 p^{2} - 50 p - 75}$

Step 1.5

Factor $5$ out of $5 (p^{2} + 4 p) + 5 \cdot 3$ .

$\frac{5 (p^{2} + 4 p + 3)}{25 p^{2} - 50 p - 75}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $5$ out of $25 p^{2}$ .

$\frac{5 (p^{2} + 4 p + 3)}{5 (5 p^{2}) - 50 p - 75}$

Step 1.6.2

Factor $5$ out of $- 50 p$ .

$\frac{5 (p^{2} + 4 p + 3)}{5 (5 p^{2}) + 5 (- 10 p) - 75}$

Step 1.6.3

Factor $5$ out of $5 (5 p^{2}) + 5 (- 10 p)$ .

$\frac{5 (p^{2} + 4 p + 3)}{5 (5 p^{2} - 10 p) - 75}$

Step 1.6.4

Factor $5$ out of $- 75$ .

$\frac{5 (p^{2} + 4 p + 3)}{5 (5 p^{2} - 10 p) + 5 (- 15)}$

Step 1.6.5

Factor $5$ out of $5 (5 p^{2} - 10 p) + 5 (- 15)$ .

$\frac{5 (p^{2} + 4 p + 3)}{5 (5 p^{2} - 10 p - 15)}$

Step 1.6.6

Cancel the common factor.

$\frac{5 (p^{2} + 4 p + 3)}{5 (5 p^{2} - 10 p - 15)}$

Step 1.6.7

Rewrite the expression.

$\frac{p^{2} + 4 p + 3}{5 p^{2} - 10 p - 15}$

Step 2

Factor $p^{2} + 4 p + 3$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(p + 1) (p + 3)}{5 p^{2} - 10 p - 15}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $5$ out of $5 p^{2} - 10 p - 15$ .

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Step 3.1.1

Factor $5$ out of $5 p^{2}$ .

$\frac{(p + 1) (p + 3)}{5 (p^{2}) - 10 p - 15}$

Step 3.1.2

Factor $5$ out of $- 10 p$ .

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

Step 3.1.3

Factor $5$ out of $- 15$ .

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

Step 3.1.4

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

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

Step 3.1.5

Factor $5$ out of $5 (p^{2} - 2 p) + 5 \cdot - 3$ .

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

Step 3.2

Factor $p^{2} - 2 p - 3$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 3.2.2

Write the factored form using these integers.

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

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

Step 4

Cancel the common factor of $p + 1$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{p + 3}{5 (p - 3)}$