Precalculus Examples

$\frac{1}{2 (v - 1)} + \frac{3}{v^{2} - 1} = \frac{1}{4}$

Step 1

Factor each term.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{2 (v - 1)} + \frac{3}{v^{2} - 1^{2}} = \frac{1}{4}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = v$ and $b = 1$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 (v - 1), (v + 1) (v - 1), 4$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 2.6

Multiply $2$ by $2$ .

$4$

Step 2.7

The factor for $v - 1$ is $v - 1$ itself.

$(v - 1) = v - 1$

$(v - 1)$ occurs $1$ time.

Step 2.8

The factor for $v + 1$ is $v + 1$ itself.

$(v + 1) = v + 1$

$(v + 1)$ occurs $1$ time.

Step 2.9

The factor for $v - 1$ is $v - 1$ itself.

$(v - 1) = v - 1$

$(v - 1)$ occurs $1$ time.

Step 2.10

The LCM of $v - 1, v + 1, v - 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(v - 1) (v + 1)$

Step 2.11

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$4 (v - 1) (v + 1)$

Step 3

Multiply each term in $\frac{1}{2 (v - 1)} + \frac{3}{(v + 1) (v - 1)} = \frac{1}{4}$ by $4 (v - 1) (v + 1)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{2 (v - 1)} + \frac{3}{(v + 1) (v - 1)} = \frac{1}{4}$ by $4 (v - 1) (v + 1)$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.2.1

Factor $2$ out of $4$ .

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

Step 3.2.1.2.2

Cancel the common factor.

$2 \cdot 2 \frac{1}{2 (v - 1)} ((v - 1) (v + 1)) + \frac{3}{(v + 1) (v - 1)} (4 (v - 1) (v + 1)) = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.1.2.3

Rewrite the expression.

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

Step 3.2.1.3

Combine $2$ and $\frac{1}{v - 1}$ .

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

Step 3.2.1.4

Cancel the common factor of $v - 1$ .

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Step 3.2.1.4.1

Cancel the common factor.

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

Step 3.2.1.4.2

Rewrite the expression.

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

Step 3.2.1.5

Apply the distributive property.

$2 v + 2 \cdot 1 + \frac{3}{(v + 1) (v - 1)} (4 (v - 1) (v + 1)) = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.1.6

Multiply $2$ by $1$ .

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

Step 3.2.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.8

Multiply $4 \frac{3}{(v + 1) (v - 1)}$ .

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Step 3.2.1.8.1

Combine $4$ and $\frac{3}{(v + 1) (v - 1)}$ .

$2 v + 2 + \frac{4 \cdot 3}{(v + 1) (v - 1)} ((v - 1) (v + 1)) = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.1.8.2

Multiply $4$ by $3$ .

$2 v + 2 + \frac{12}{(v + 1) (v - 1)} ((v - 1) (v + 1)) = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.1.9

Cancel the common factor of $(v - 1) (v + 1)$ .

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Step 3.2.1.9.1

Factor $(v - 1) (v + 1)$ out of $(v + 1) (v - 1)$ .

$2 v + 2 + \frac{12}{(v - 1) (v + 1) (1)} ((v - 1) (v + 1)) = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.1.9.2

Cancel the common factor.

$2 v + 2 + \frac{12}{(v - 1) (v + 1) \cdot 1} ((v - 1) (v + 1)) = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.1.9.3

Rewrite the expression.

$2 v + 2 + 12 = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.2.2

Add $2$ and $12$ .

$2 v + 14 = \frac{1}{4} (4 (v - 1) (v + 1))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $4$ .

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Step 3.3.1.1

Factor $4$ out of $4 (v - 1) (v + 1)$ .

$2 v + 14 = \frac{1}{4} (4 ((v - 1) (v + 1)))$

Step 3.3.1.2

Cancel the common factor.

$2 v + 14 = \frac{1}{4} (4 ((v - 1) (v + 1)))$

Step 3.3.1.3

Rewrite the expression.

$2 v + 14 = (v - 1) (v + 1)$

Step 3.3.2

Expand $(v - 1) (v + 1)$ using the FOIL Method.

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Step 3.3.2.1

Apply the distributive property.

$2 v + 14 = v (v + 1) - 1 (v + 1)$

Step 3.3.2.2

Apply the distributive property.

$2 v + 14 = v \cdot v + v \cdot 1 - 1 (v + 1)$

Step 3.3.2.3

Apply the distributive property.

$2 v + 14 = v \cdot v + v \cdot 1 - 1 v - 1 \cdot 1$

Step 3.3.3

Simplify terms.

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Step 3.3.3.1

Combine the opposite terms in $v \cdot v + v \cdot 1 - 1 v - 1 \cdot 1$ .

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Step 3.3.3.1.1

Reorder the factors in the terms $v \cdot 1$ and $- 1 v$ .

$2 v + 14 = v \cdot v + 1 v - 1 v - 1 \cdot 1$

Step 3.3.3.1.2

Subtract $1 v$ from $1 v$ .

$2 v + 14 = v \cdot v + 0 - 1 \cdot 1$

Step 3.3.3.1.3

Add $v \cdot v$ and $0$ .

$2 v + 14 = v \cdot v - 1 \cdot 1$

Step 3.3.3.2

Simplify each term.

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Step 3.3.3.2.1

Multiply $v$ by $v$ .

$2 v + 14 = v^{2} - 1 \cdot 1$

Step 3.3.3.2.2

Multiply $- 1$ by $1$ .

$2 v + 14 = v^{2} - 1$

Step 4

Solve the equation.

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Step 4.1

Subtract $v^{2}$ from both sides of the equation.

$2 v + 14 - v^{2} = - 1$

Step 4.2

Add $1$ to both sides of the equation.

$2 v + 14 - v^{2} + 1 = 0$

Step 4.3

Add $14$ and $1$ .

$2 v - v^{2} + 15 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $- 1$ out of $2 v - v^{2} + 15$ .

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Step 4.4.1.1

Reorder $2 v$ and $- v^{2}$ .

$- v^{2} + 2 v + 15 = 0$

Step 4.4.1.2

Factor $- 1$ out of $- v^{2}$ .

$- (v^{2}) + 2 v + 15 = 0$

Step 4.4.1.3

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

$- (v^{2}) - (- 2 v) + 15 = 0$

Step 4.4.1.4

Rewrite $15$ as $- 1 (- 15)$ .

$- (v^{2}) - (- 2 v) - 1 \cdot - 15 = 0$

Step 4.4.1.5

Factor $- 1$ out of $- (v^{2}) - (- 2 v)$ .

$- (v^{2} - 2 v) - 1 \cdot - 15 = 0$

Step 4.4.1.6

Factor $- 1$ out of $- (v^{2} - 2 v) - 1 (- 15)$ .

$- (v^{2} - 2 v - 15) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor $v^{2} - 2 v - 15$ using the AC method.

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Step 4.4.2.1.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 4.4.2.1.2

Write the factored form using these integers.

$- ((v - 5) (v + 3)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$- (v - 5) (v + 3) = 0$

Step 4.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$v - 5 = 0$

$v + 3 = 0$

Step 4.6

Set $v - 5$ equal to $0$ and solve for $v$ .

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Step 4.6.1

Set $v - 5$ equal to $0$ .

$v - 5 = 0$

Step 4.6.2

Add $5$ to both sides of the equation.

$v = 5$

Step 4.7

Set $v + 3$ equal to $0$ and solve for $v$ .

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Step 4.7.1

Set $v + 3$ equal to $0$ .

$v + 3 = 0$

Step 4.7.2

Subtract $3$ from both sides of the equation.

$v = - 3$

Step 4.8

The final solution is all the values that make $- (v - 5) (v + 3) = 0$ true.

$v = 5, - 3$