Algebra Examples

$\frac{- 3 d}{d^{2} - 2 d - 8} + \frac{3}{d - 4} = \frac{- 2}{d + 2}$

Step 1

Subtract $\frac{- 2}{d + 2}$ from both sides of the equation.

$\frac{- 3 d}{d^{2} - 2 d - 8} + \frac{3}{d - 4} - \frac{- 2}{d + 2} = 0$

Step 2

Simplify $\frac{- 3 d}{d^{2} - 2 d - 8} + \frac{3}{d - 4} - \frac{- 2}{d + 2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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 $- 2$ .

$- 4, 2$

Step 2.1.1.2

Write the factored form using these integers.

$\frac{- 3 d}{(d - 4) (d + 2)} + \frac{3}{d - 4} - \frac{- 2}{d + 2} = 0$

Step 2.1.2

Move the negative in front of the fraction.

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3}{d - 4} - \frac{- 2}{d + 2} = 0$

Step 2.1.3

Move the negative in front of the fraction.

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3}{d - 4} - - \frac{2}{d + 2} = 0$

Step 2.1.4

Multiply $- - \frac{2}{d + 2}$ .

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Step 2.1.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3}{d - 4} + 1 \frac{2}{d + 2} = 0$

Step 2.1.4.2

Multiply $\frac{2}{d + 2}$ by $1$ .

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3}{d - 4} + \frac{2}{d + 2} = 0$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{3}{d - 4}$ by $\frac{d + 2}{d + 2}$ .

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3}{d - 4} \cdot \frac{d + 2}{d + 2} + \frac{2}{d + 2} = 0$

Step 2.2.2

Multiply $\frac{3}{d - 4}$ by $\frac{d + 2}{d + 2}$ .

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3 (d + 2)}{(d - 4) (d + 2)} + \frac{2}{d + 2} = 0$

Step 2.2.3

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

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3 (d + 2)}{(d - 4) (d + 2)} + \frac{2}{d + 2} \cdot \frac{d - 4}{d - 4} = 0$

Step 2.2.4

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

$- \frac{3 d}{(d - 4) (d + 2)} + \frac{3 (d + 2)}{(d - 4) (d + 2)} + \frac{2 (d - 4)}{(d + 2) (d - 4)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{- 3 d + 3 (d + 2) + 2 (d - 4)}{(d + 2) (d - 4)} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

$\frac{- 3 d + 3 d + 3 \cdot 2 + 2 (d - 4)}{(d + 2) (d - 4)} = 0$

Step 2.4.2

Multiply $3$ by $2$ .

$\frac{- 3 d + 3 d + 6 + 2 (d - 4)}{(d + 2) (d - 4)} = 0$

Step 2.4.3

Apply the distributive property.

$\frac{- 3 d + 3 d + 6 + 2 d + 2 \cdot - 4}{(d + 2) (d - 4)} = 0$

Step 2.4.4

Multiply $2$ by $- 4$ .

$\frac{- 3 d + 3 d + 6 + 2 d - 8}{(d + 2) (d - 4)} = 0$

Step 2.5

Add $- 3 d$ and $3 d$ .

$\frac{0 + 6 + 2 d - 8}{(d + 2) (d - 4)} = 0$

Step 2.6

Add $0$ and $6$ .

$\frac{6 + 2 d - 8}{(d + 2) (d - 4)} = 0$

Step 2.7

Subtract $8$ from $6$ .

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

Step 2.8

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

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Step 2.8.1

Factor $2$ out of $2 d$ .

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

Step 2.8.2

Factor $2$ out of $- 2$ .

$\frac{2 (d) + 2 (- 1)}{(d + 2) (d - 4)} = 0$

Step 2.8.3

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

$\frac{2 (d - 1)}{(d + 2) (d - 4)} = 0$

Step 3

Set the numerator equal to zero.

$2 (d - 1) = 0$

Step 4

Solve the equation for $d$ .

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Step 4.1

Divide each term in $2 (d - 1) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (d - 1) = 0$ by $2$ .

$\frac{2 (d - 1)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (d - 1)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $d - 1$ by $1$ .

$d - 1 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$d - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$d = 1$