Trigonometry Examples

$\frac{d^{2} + d - 30}{d^{2} + 3 d - 40} + \frac{d^{2} + 14 d + 48}{d^{2} - 2 d - 48}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor $d^{2} + d - 30$ using the AC method.

Tap for more steps...

Step 1.1.1

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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 1.1.2

Write the factored form using these integers.

$\frac{(d - 5) (d + 6)}{d^{2} + 3 d - 40} + \frac{d^{2} + 14 d + 48}{d^{2} - 2 d - 48}$

Step 1.2

Factor $d^{2} + 3 d - 40$ using the AC method.

Tap for more steps...

Step 1.2.1

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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Step 1.2.2

Write the factored form using these integers.

$\frac{(d - 5) (d + 6)}{(d - 5) (d + 8)} + \frac{d^{2} + 14 d + 48}{d^{2} - 2 d - 48}$

Step 1.3

Cancel the common factor of $d - 5$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\frac{(d - 5) (d + 6)}{(d - 5) (d + 8)} + \frac{d^{2} + 14 d + 48}{d^{2} - 2 d - 48}$

Step 1.3.2

Rewrite the expression.

$\frac{d + 6}{d + 8} + \frac{d^{2} + 14 d + 48}{d^{2} - 2 d - 48}$

Step 1.4

Factor $d^{2} + 14 d + 48$ using the AC method.

Tap for more steps...

Step 1.4.1

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 $48$ and whose sum is $14$ .

$6, 8$

Step 1.4.2

Write the factored form using these integers.

$\frac{d + 6}{d + 8} + \frac{(d + 6) (d + 8)}{d^{2} - 2 d - 48}$

Step 1.5

Factor $d^{2} - 2 d - 48$ using the AC method.

Tap for more steps...

Step 1.5.1

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 $- 48$ and whose sum is $- 2$ .

$- 8, 6$

Step 1.5.2

Write the factored form using these integers.

$\frac{d + 6}{d + 8} + \frac{(d + 6) (d + 8)}{(d - 8) (d + 6)}$

Step 1.6

Cancel the common factor of $d + 6$ .

Tap for more steps...

Step 1.6.1

Cancel the common factor.

$\frac{d + 6}{d + 8} + \frac{(d + 6) (d + 8)}{(d - 8) (d + 6)}$

Step 1.6.2

Rewrite the expression.

$\frac{d + 6}{d + 8} + \frac{d + 8}{d - 8}$

Step 2

To write $\frac{d + 6}{d + 8}$ as a fraction with a common denominator, multiply by $\frac{d - 8}{d - 8}$ .

$\frac{d + 6}{d + 8} \cdot \frac{d - 8}{d - 8} + \frac{d + 8}{d - 8}$

Step 3

To write $\frac{d + 8}{d - 8}$ as a fraction with a common denominator, multiply by $\frac{d + 8}{d + 8}$ .

$\frac{d + 6}{d + 8} \cdot \frac{d - 8}{d - 8} + \frac{d + 8}{d - 8} \cdot \frac{d + 8}{d + 8}$

Step 4

Write each expression with a common denominator of $(d + 8) (d - 8)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{d + 6}{d + 8}$ by $\frac{d - 8}{d - 8}$ .

$\frac{(d + 6) (d - 8)}{(d + 8) (d - 8)} + \frac{d + 8}{d - 8} \cdot \frac{d + 8}{d + 8}$

Step 4.2

Multiply $\frac{d + 8}{d - 8}$ by $\frac{d + 8}{d + 8}$ .

$\frac{(d + 6) (d - 8)}{(d + 8) (d - 8)} + \frac{(d + 8) (d + 8)}{(d - 8) (d + 8)}$

Step 4.3

Reorder the factors of $(d - 8) (d + 8)$ .

$\frac{(d + 6) (d - 8)}{(d + 8) (d - 8)} + \frac{(d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(d + 6) (d - 8) + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Expand $(d + 6) (d - 8)$ using the FOIL Method.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

$\frac{d (d - 8) + 6 (d - 8) + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.1.2

Apply the distributive property.

$\frac{d \cdot d + d \cdot - 8 + 6 (d - 8) + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.1.3

Apply the distributive property.

$\frac{d \cdot d + d \cdot - 8 + 6 d + 6 \cdot - 8 + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.2

Simplify and combine like terms.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $d$ by $d$ .

$\frac{d^{2} + d \cdot - 8 + 6 d + 6 \cdot - 8 + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.2.1.2

Move $- 8$ to the left of $d$ .

$\frac{d^{2} - 8 \cdot d + 6 d + 6 \cdot - 8 + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.2.1.3

Multiply $6$ by $- 8$ .

$\frac{d^{2} - 8 d + 6 d - 48 + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.2.2

Add $- 8 d$ and $6 d$ .

$\frac{d^{2} - 2 d - 48 + (d + 8) (d + 8)}{(d + 8) (d - 8)}$

Step 6.3

Expand $(d + 8) (d + 8)$ using the FOIL Method.

Tap for more steps...

Step 6.3.1

Apply the distributive property.

$\frac{d^{2} - 2 d - 48 + d (d + 8) + 8 (d + 8)}{(d + 8) (d - 8)}$

Step 6.3.2

Apply the distributive property.

$\frac{d^{2} - 2 d - 48 + d \cdot d + d \cdot 8 + 8 (d + 8)}{(d + 8) (d - 8)}$

Step 6.3.3

Apply the distributive property.

$\frac{d^{2} - 2 d - 48 + d \cdot d + d \cdot 8 + 8 d + 8 \cdot 8}{(d + 8) (d - 8)}$

Step 6.4

Simplify and combine like terms.

Tap for more steps...

Step 6.4.1

Simplify each term.

Tap for more steps...

Step 6.4.1.1

Multiply $d$ by $d$ .

$\frac{d^{2} - 2 d - 48 + d^{2} + d \cdot 8 + 8 d + 8 \cdot 8}{(d + 8) (d - 8)}$

Step 6.4.1.2

Move $8$ to the left of $d$ .

$\frac{d^{2} - 2 d - 48 + d^{2} + 8 \cdot d + 8 d + 8 \cdot 8}{(d + 8) (d - 8)}$

Step 6.4.1.3

Multiply $8$ by $8$ .

$\frac{d^{2} - 2 d - 48 + d^{2} + 8 d + 8 d + 64}{(d + 8) (d - 8)}$

Step 6.4.2

Add $8 d$ and $8 d$ .

$\frac{d^{2} - 2 d - 48 + d^{2} + 16 d + 64}{(d + 8) (d - 8)}$

Step 6.5

Add $d^{2}$ and $d^{2}$ .

$\frac{2 d^{2} - 2 d - 48 + 16 d + 64}{(d + 8) (d - 8)}$

Step 6.6

Add $- 2 d$ and $16 d$ .

$\frac{2 d^{2} + 14 d - 48 + 64}{(d + 8) (d - 8)}$

Step 6.7

Add $- 48$ and $64$ .

$\frac{2 d^{2} + 14 d + 16}{(d + 8) (d - 8)}$

Step 6.8

Factor $2$ out of $2 d^{2} + 14 d + 16$ .

Tap for more steps...

Step 6.8.1

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

$\frac{2 (d^{2}) + 14 d + 16}{(d + 8) (d - 8)}$

Step 6.8.2

Factor $2$ out of $14 d$ .

$\frac{2 (d^{2}) + 2 (7 d) + 16}{(d + 8) (d - 8)}$

Step 6.8.3

Factor $2$ out of $16$ .

$\frac{2 d^{2} + 2 (7 d) + 2 \cdot 8}{(d + 8) (d - 8)}$

Step 6.8.4

Factor $2$ out of $2 d^{2} + 2 (7 d)$ .

$\frac{2 (d^{2} + 7 d) + 2 \cdot 8}{(d + 8) (d - 8)}$

Step 6.8.5

Factor $2$ out of $2 (d^{2} + 7 d) + 2 \cdot 8$ .

$\frac{2 (d^{2} + 7 d + 8)}{(d + 8) (d - 8)}$