Algebra Examples

$\frac{x - 1}{8 x} = \frac{x^{2} + 6 x + 5}{8 x^{2}} - \frac{1}{x}$

Step 1

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

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Step 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 $5$ and whose sum is $6$ .

$1, 5$

Step 1.2

Write the factored form using these integers.

$\frac{x - 1}{8 x} = \frac{(x + 1) (x + 5)}{8 x^{2}} - \frac{1}{x}$

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.

$8 x, 8 x^{2}, x$

Step 2.2

Since $8 x, 8 x^{2}, x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $8, 8, 1$ then find LCM for the variable part $x^{1}, x^{2}, x^{1}$ .

Step 2.3

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 2.4

The prime factors for $8$ are $2 \cdot 2 \cdot 2$ .

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Step 2.4.1

$8$ has factors of $2$ and $4$ .

$2 \cdot 4$

Step 2.4.2

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

$2 \cdot 2 \cdot 2$

Step 2.5

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

Not prime

Step 2.6

The LCM of $8, 8, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 2$

Step 2.7

Multiply $2 \cdot 2 \cdot 2$ .

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Step 2.7.1

Multiply $2$ by $2$ .

$4 \cdot 2$

Step 2.7.2

Multiply $4$ by $2$ .

$8$

Step 2.8

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 2.9

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 2.10

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 2.11

The LCM of $x^{1}, x^{2}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 2.12

Multiply $x$ by $x$ .

$x^{2}$

Step 2.13

The LCM for $8 x, 8 x^{2}, x$ is the numeric part $8$ multiplied by the variable part.

$8 x^{2}$

Step 3

Multiply each term in $\frac{x - 1}{8 x} = \frac{(x + 1) (x + 5)}{8 x^{2}} - \frac{1}{x}$ by $8 x^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{x - 1}{8 x} = \frac{(x + 1) (x + 5)}{8 x^{2}} - \frac{1}{x}$ by $8 x^{2}$ .

$\frac{x - 1}{8 x} (8 x^{2}) = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$8 \frac{x - 1}{8 x} x^{2} = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.2

Cancel the common factor of $8$ .

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Step 3.2.2.1

Factor $8$ out of $8 x$ .

$8 \frac{x - 1}{8 (x)} x^{2} = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.2.2

Cancel the common factor.

$8 \frac{x - 1}{8 x} x^{2} = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.2.3

Rewrite the expression.

$\frac{x - 1}{x} x^{2} = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.3

Cancel the common factor of $x$ .

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Step 3.2.3.1

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

$\frac{x - 1}{x} (x \cdot x) = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.3.2

Cancel the common factor.

$\frac{x - 1}{x} (x \cdot x) = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.3.3

Rewrite the expression.

$(x - 1) x = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.4

Apply the distributive property.

$x \cdot x - 1 x = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Multiply $x$ by $x$ .

$x^{2} - 1 x = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.2.5.2

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x = \frac{(x + 1) (x + 5)}{8 x^{2}} (8 x^{2}) - \frac{1}{x} (8 x^{2})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$x^{2} - x = 8 \frac{(x + 1) (x + 5)}{8 x^{2}} x^{2} - \frac{1}{x} (8 x^{2})$

Step 3.3.1.2

Cancel the common factor of $8$ .

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Step 3.3.1.2.1

Factor $8$ out of $8 x^{2}$ .

$x^{2} - x = 8 \frac{(x + 1) (x + 5)}{8 (x^{2})} x^{2} - \frac{1}{x} (8 x^{2})$

Step 3.3.1.2.2

Cancel the common factor.

$x^{2} - x = 8 \frac{(x + 1) (x + 5)}{8 x^{2}} x^{2} - \frac{1}{x} (8 x^{2})$

Step 3.3.1.2.3

Rewrite the expression.

$x^{2} - x = \frac{(x + 1) (x + 5)}{x^{2}} x^{2} - \frac{1}{x} (8 x^{2})$

Step 3.3.1.3

Cancel the common factor of $x^{2}$ .

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Step 3.3.1.3.1

Cancel the common factor.

$x^{2} - x = \frac{(x + 1) (x + 5)}{x^{2}} x^{2} - \frac{1}{x} (8 x^{2})$

Step 3.3.1.3.2

Rewrite the expression.

$x^{2} - x = (x + 1) (x + 5) - \frac{1}{x} (8 x^{2})$

Step 3.3.1.4

Expand $(x + 1) (x + 5)$ using the FOIL Method.

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Step 3.3.1.4.1

Apply the distributive property.

$x^{2} - x = x (x + 5) + 1 (x + 5) - \frac{1}{x} (8 x^{2})$

Step 3.3.1.4.2

Apply the distributive property.

$x^{2} - x = x \cdot x + x \cdot 5 + 1 (x + 5) - \frac{1}{x} (8 x^{2})$

Step 3.3.1.4.3

Apply the distributive property.

$x^{2} - x = x \cdot x + x \cdot 5 + 1 x + 1 \cdot 5 - \frac{1}{x} (8 x^{2})$

Step 3.3.1.5

Simplify and combine like terms.

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Step 3.3.1.5.1

Simplify each term.

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Step 3.3.1.5.1.1

Multiply $x$ by $x$ .

$x^{2} - x = x^{2} + x \cdot 5 + 1 x + 1 \cdot 5 - \frac{1}{x} (8 x^{2})$

Step 3.3.1.5.1.2

Move $5$ to the left of $x$ .

$x^{2} - x = x^{2} + 5 \cdot x + 1 x + 1 \cdot 5 - \frac{1}{x} (8 x^{2})$

Step 3.3.1.5.1.3

Multiply $x$ by $1$ .

$x^{2} - x = x^{2} + 5 x + x + 1 \cdot 5 - \frac{1}{x} (8 x^{2})$

Step 3.3.1.5.1.4

Multiply $5$ by $1$ .

$x^{2} - x = x^{2} + 5 x + x + 5 - \frac{1}{x} (8 x^{2})$

Step 3.3.1.5.2

Add $5 x$ and $x$ .

$x^{2} - x = x^{2} + 6 x + 5 - \frac{1}{x} (8 x^{2})$

Step 3.3.1.6

Cancel the common factor of $x$ .

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Step 3.3.1.6.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$x^{2} - x = x^{2} + 6 x + 5 + \frac{- 1}{x} (8 x^{2})$

Step 3.3.1.6.2

Factor $x$ out of $8 x^{2}$ .

$x^{2} - x = x^{2} + 6 x + 5 + \frac{- 1}{x} (x (8 x))$

Step 3.3.1.6.3

Cancel the common factor.

$x^{2} - x = x^{2} + 6 x + 5 + \frac{- 1}{x} (x (8 x))$

Step 3.3.1.6.4

Rewrite the expression.

$x^{2} - x = x^{2} + 6 x + 5 - (8 x)$

Step 3.3.1.7

Multiply $8$ by $- 1$ .

$x^{2} - x = x^{2} + 6 x + 5 - 8 x$

Step 3.3.2

Subtract $8 x$ from $6 x$ .

$x^{2} - x = x^{2} - 2 x + 5$

Step 4

Move all terms containing $x$ to the left side of the equation.

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Step 4.1

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

$x^{2} - x - x^{2} = - 2 x + 5$

Step 4.2

Add $2 x$ to both sides of the equation.

$x^{2} - x - x^{2} + 2 x = 5$

Step 4.3

Combine the opposite terms in $x^{2} - x - x^{2} + 2 x$ .

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Step 4.3.1

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

$- x + 0 + 2 x = 5$

Step 4.3.2

Add $- x$ and $0$ .

$- x + 2 x = 5$

Step 4.4

Add $- x$ and $2 x$ .

$x = 5$