Algebra Examples

$\frac{5 - x}{x^{2} - 3 x - 10} = 2$

Step 1

Factor each term.

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Step 1.1

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Cancel the common factor of $5 - x$ and $x - 5$ .

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Step 1.2.1

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

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

Step 1.2.2

Factor $- 1$ out of $- x$ .

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

Step 1.2.3

Factor $- 1$ out of $- 1 (- 5) - (x)$ .

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

Step 1.2.4

Reorder terms.

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

Step 1.2.5

Cancel the common factor.

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

Step 1.2.6

Rewrite the expression.

$\frac{- 1}{x + 2} = 2$

Step 1.3

Move the negative in front of the fraction.

$- \frac{1}{x + 2} = 2$

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.

$x + 2, 1$

Step 2.2

Remove parentheses.

$x + 2, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x + 2$

Step 3

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

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Step 3.1

Multiply each term in $- \frac{1}{x + 2} = 2$ by $x + 2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x + 2$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{1}{x + 2}$ into the numerator.

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

$- 1 = 2 (x + 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$- 1 = 2 x + 2 \cdot 2$

Step 3.3.2

Multiply $2$ by $2$ .

$- 1 = 2 x + 4$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $2 x + 4 = - 1$ .

$2 x + 4 = - 1$

Step 4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$2 x = - 1 - 4$

Step 4.2.2

Subtract $4$ from $- 1$ .

$2 x = - 5$

Step 4.3

Divide each term in $2 x = - 5$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{2}$

Decimal Form:

$x = - 2.5$

Mixed Number Form:

$x = - 2 \frac{1}{2}$