Trigonometry Examples

$\frac{h - 2}{h^{2} + 4 h + 4} + \frac{h - 2}{h + 2}$

Step 1

To write $\frac{h - 2}{h^{2} + 4 h + 4}$ as a fraction with a common denominator, multiply by $\frac{h + 2}{h + 2}$ .

$\frac{h - 2}{h^{2} + 4 h + 4} \cdot \frac{h + 2}{h + 2} + \frac{h - 2}{h + 2}$

Step 2

To write $\frac{h - 2}{h + 2}$ as a fraction with a common denominator, multiply by $\frac{h^{2} + 4 h + 4}{h^{2} + 4 h + 4}$ .

$\frac{h - 2}{h^{2} + 4 h + 4} \cdot \frac{h + 2}{h + 2} + \frac{h - 2}{h + 2} \cdot \frac{h^{2} + 4 h + 4}{h^{2} + 4 h + 4}$

Step 3

Write each expression with a common denominator of $(h^{2} + 4 h + 4) (h + 2)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{h - 2}{h^{2} + 4 h + 4}$ by $\frac{h + 2}{h + 2}$ .

$\frac{(h - 2) (h + 2)}{(h^{2} + 4 h + 4) (h + 2)} + \frac{h - 2}{h + 2} \cdot \frac{h^{2} + 4 h + 4}{h^{2} + 4 h + 4}$

Step 3.2

Multiply $\frac{h - 2}{h + 2}$ by $\frac{h^{2} + 4 h + 4}{h^{2} + 4 h + 4}$ .

$\frac{(h - 2) (h + 2)}{(h^{2} + 4 h + 4) (h + 2)} + \frac{(h - 2) (h^{2} + 4 h + 4)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 3.3

Reorder the factors of $(h^{2} + 4 h + 4) (h + 2)$ .

$\frac{(h - 2) (h + 2)}{(h + 2) (h^{2} + 4 h + 4)} + \frac{(h - 2) (h^{2} + 4 h + 4)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(h - 2) (h + 2) + (h - 2) (h^{2} + 4 h + 4)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5

Rewrite $\frac{(h - 2) (h + 2) + (h - 2) (h^{2} + 4 h + 4)}{(h + 2) (h^{2} + 4 h + 4)}$ in a factored form.

Tap for more steps...

Step 5.1

Factor $h - 2$ out of $(h - 2) (h + 2) + (h - 2) (h^{2} + 4 h + 4)$ .

$\frac{(h - 2) (h + 2 + h^{2} + 4 h + 4)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5.2

Add $h$ and $4 h$ .

$\frac{(h - 2) (5 h + 2 + h^{2} + 4)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5.3

Add $2$ and $4$ .

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

Step 5.4

Let $u = h$ . Substitute $u$ for all occurrences of $h$ .

$\frac{(h - 2) (5 u + u^{2} + 6)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5.5

Factor $5 u + u^{2} + 6$ using the AC method.

Tap for more steps...

Step 5.5.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 5.5.2

Write the factored form using these integers.

$\frac{(h - 2) ((u + 2) (u + 3))}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5.6

Factor.

Tap for more steps...

Step 5.6.1

Replace all occurrences of $u$ with $h$ .

$\frac{(h - 2) ((h + 2) (h + 3))}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5.6.2

Remove unnecessary parentheses.

$\frac{(h - 2) (h + 2) (h + 3)}{(h + 2) (h^{2} + 4 h + 4)}$

Step 5.7

Factor using the perfect square rule.

Tap for more steps...

Step 5.7.1

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

$\frac{(h - 2) (h + 2) (h + 3)}{(h + 2) (h^{2} + 4 h + 2^{2})}$

Step 5.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 h = 2 \cdot h \cdot 2$

Step 5.7.3

Rewrite the polynomial.

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

Step 5.7.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = h$ and $b = 2$ .

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

Step 5.8

Reduce the expression $\frac{(h - 2) (h + 2) (h + 3)}{(h + 2) {(h + 2)}^{2}}$ by cancelling the common factors.

Tap for more steps...

Step 5.8.1

Cancel the common factor.

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

Step 5.8.2

Rewrite the expression.

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