Trigonometry Examples

$\frac{\frac{p^{2} + 20 + 100}{p^{2} - 5 p - 50}}{\frac{p^{2} - 100}{7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{p^{2} + 20 + 100}{p^{2} - 5 p - 50} \cdot \frac{7}{p^{2} - 100}$

Step 2

Add $20$ and $100$ .

$\frac{p^{2} + 120}{p^{2} - 5 p - 50} \cdot \frac{7}{p^{2} - 100}$

Step 3

Factor $p^{2} - 5 p - 50$ using the AC method.

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Step 3.1

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 $- 50$ and whose sum is $- 5$ .

$- 10, 5$

Step 3.2

Write the factored form using these integers.

$\frac{p^{2} + 120}{(p - 10) (p + 5)} \cdot \frac{7}{p^{2} - 100}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $100$ as $10^{2}$ .

$\frac{p^{2} + 120}{(p - 10) (p + 5)} \cdot \frac{7}{p^{2} - 10^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = p$ and $b = 10$ .

$\frac{p^{2} + 120}{(p - 10) (p + 5)} \cdot \frac{7}{(p + 10) (p - 10)}$

Step 5

Multiply $\frac{p^{2} + 120}{(p - 10) (p + 5)} \cdot \frac{7}{(p + 10) (p - 10)}$ .

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Step 5.1

Multiply $\frac{p^{2} + 120}{(p - 10) (p + 5)}$ by $\frac{7}{(p + 10) (p - 10)}$ .

$\frac{(p^{2} + 120) \cdot 7}{(p - 10) (p + 5) ((p + 10) (p - 10))}$

Step 5.2

Raise $p - 10$ to the power of $1$ .

$\frac{(p^{2} + 120) \cdot 7}{{(p - 10)}^{1} (p - 10) (p + 5) (p + 10)}$

Step 5.3

Raise $p - 10$ to the power of $1$ .

$\frac{(p^{2} + 120) \cdot 7}{{(p - 10)}^{1} {(p - 10)}^{1} (p + 5) (p + 10)}$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(p^{2} + 120) \cdot 7}{{(p - 10)}^{1 + 1} (p + 5) (p + 10)}$

Step 5.5

Add $1$ and $1$ .

$\frac{(p^{2} + 120) \cdot 7}{{(p - 10)}^{2} (p + 5) (p + 10)}$

Step 6

Move $7$ to the left of $p^{2} + 120$ .

$\frac{7 (p^{2} + 120)}{{(p - 10)}^{2} (p + 5) (p + 10)}$