Algebra Examples

$\frac{d - 4}{d^{2} + 2 d - 8} - \frac{d + 2}{d^{2} - 16}$

Step 1

Simplify each term.

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Step 1.1

Factor $d^{2} + 2 d - 8$ using the AC method.

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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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{d - 4}{(d - 2) (d + 4)} - \frac{d + 2}{d^{2} - 16}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = d$ and $b = 4$ .

$\frac{d - 4}{(d - 2) (d + 4)} - \frac{d + 2}{(d + 4) (d - 4)}$

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(d - 2) (d + 4) (d - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{d - 4}{(d - 2) (d + 4)}$ by $\frac{d - 4}{d - 4}$ .

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

Step 4.2

Multiply $\frac{d + 2}{(d + 4) (d - 4)}$ by $\frac{d - 2}{d - 2}$ .

$\frac{(d - 4) (d - 4)}{(d - 2) (d + 4) (d - 4)} - \frac{(d + 2) (d - 2)}{(d + 4) (d - 4) (d - 2)}$

Step 4.3

Reorder the factors of $(d - 2) (d + 4) (d - 4)$ .

$\frac{(d - 4) (d - 4)}{(d + 4) (d - 2) (d - 4)} - \frac{(d + 2) (d - 2)}{(d + 4) (d - 4) (d - 2)}$

Step 4.4

Reorder the factors of $(d + 4) (d - 4) (d - 2)$ .

$\frac{(d - 4) (d - 4)}{(d + 4) (d - 2) (d - 4)} - \frac{(d + 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(d - 4) (d - 4) - (d + 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(d - 4) (d - 4)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{d (d - 4) - 4 (d - 4) - (d + 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $d$ by $d$ .

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

Step 6.2.1.2

Move $- 4$ to the left of $d$ .

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

Step 6.2.1.3

Multiply $- 4$ by $- 4$ .

$\frac{d^{2} - 4 d - 4 d + 16 - (d + 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.2.2

Subtract $4 d$ from $- 4 d$ .

$\frac{d^{2} - 8 d + 16 - (d + 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.3

Apply the distributive property.

$\frac{d^{2} - 8 d + 16 + (- d - 1 \cdot 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.4

Multiply $- 1$ by $2$ .

$\frac{d^{2} - 8 d + 16 + (- d - 2) (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.5

Expand $(- d - 2) (d - 2)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{d^{2} - 8 d + 16 - d (d - 2) - 2 (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.5.2

Apply the distributive property.

$\frac{d^{2} - 8 d + 16 - d \cdot d - d \cdot - 2 - 2 (d - 2)}{(d + 4) (d - 2) (d - 4)}$

Step 6.5.3

Apply the distributive property.

$\frac{d^{2} - 8 d + 16 - d \cdot d - d \cdot - 2 - 2 d - 2 \cdot - 2}{(d + 4) (d - 2) (d - 4)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Multiply $d$ by $d$ by adding the exponents.

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Step 6.6.1.1.1

Move $d$ .

$\frac{d^{2} - 8 d + 16 - (d \cdot d) - d \cdot - 2 - 2 d - 2 \cdot - 2}{(d + 4) (d - 2) (d - 4)}$

Step 6.6.1.1.2

Multiply $d$ by $d$ .

$\frac{d^{2} - 8 d + 16 - d^{2} - d \cdot - 2 - 2 d - 2 \cdot - 2}{(d + 4) (d - 2) (d - 4)}$

Step 6.6.1.2

Multiply $- 2$ by $- 1$ .

$\frac{d^{2} - 8 d + 16 - d^{2} + 2 d - 2 d - 2 \cdot - 2}{(d + 4) (d - 2) (d - 4)}$

Step 6.6.1.3

Multiply $- 2$ by $- 2$ .

$\frac{d^{2} - 8 d + 16 - d^{2} + 2 d - 2 d + 4}{(d + 4) (d - 2) (d - 4)}$

Step 6.6.2

Subtract $2 d$ from $2 d$ .

$\frac{d^{2} - 8 d + 16 - d^{2} + 0 + 4}{(d + 4) (d - 2) (d - 4)}$

Step 6.6.3

Add $- d^{2}$ and $0$ .

$\frac{d^{2} - 8 d + 16 - d^{2} + 4}{(d + 4) (d - 2) (d - 4)}$

Step 6.7

Subtract $d^{2}$ from $d^{2}$ .

$\frac{- 8 d + 16 + 0 + 4}{(d + 4) (d - 2) (d - 4)}$

Step 6.8

Add $- 8 d$ and $0$ .

$\frac{- 8 d + 16 + 4}{(d + 4) (d - 2) (d - 4)}$

Step 6.9

Add $16$ and $4$ .

$\frac{- 8 d + 20}{(d + 4) (d - 2) (d - 4)}$

Step 6.10

Factor $4$ out of $- 8 d + 20$ .

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Step 6.10.1

Factor $4$ out of $- 8 d$ .

$\frac{4 (- 2 d) + 20}{(d + 4) (d - 2) (d - 4)}$

Step 6.10.2

Factor $4$ out of $20$ .

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

Step 6.10.3

Factor $4$ out of $4 (- 2 d) + 4 (5)$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 2 d$ .

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

Step 7.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{4 (- (2 d) - 1 (- 5))}{(d + 4) (d - 2) (d - 4)}$

Step 7.3

Factor $- 1$ out of $- (2 d) - 1 (- 5)$ .

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

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (2 d - 5)$ as $- 1 (2 d - 5)$ .

$\frac{4 (- 1 (2 d - 5))}{(d + 4) (d - 2) (d - 4)}$

Step 7.4.2

Move the negative in front of the fraction.

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