Algebra Examples

$x - 1 = \frac{x^{2} - 4 x + 3}{x + 2}$

Step 1

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

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Step 1.1

Add $1$ to both sides of the equation.

$x = \frac{x^{2} - 4 x + 3}{x + 2} + 1$

Step 1.2

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 1.2.1

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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 1.2.2

Write the factored form using these integers.

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

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.

$1, x + 2, 1$

Step 2.2

Remove parentheses.

$1, x + 2, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x + 2$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 = \frac{(x - 3) (x - 1)}{x + 2} (x + 2) + 1 (x + 2)$

Step 3.2.2.2

Move $2$ to the left of $x$ .

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

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

Cancel the common factor of $x + 2$ .

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Step 3.3.1.1.1

Cancel the common factor.

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

Step 3.3.1.1.2

Rewrite the expression.

$x^{2} + 2 x = (x - 3) (x - 1) + 1 (x + 2)$

Step 3.3.1.2

Expand $(x - 3) (x - 1)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x^{2} + 2 x = x (x - 1) - 3 (x - 1) + 1 (x + 2)$

Step 3.3.1.2.2

Apply the distributive property.

$x^{2} + 2 x = x \cdot x + x \cdot - 1 - 3 (x - 1) + 1 (x + 2)$

Step 3.3.1.2.3

Apply the distributive property.

$x^{2} + 2 x = x \cdot x + x \cdot - 1 - 3 x - 3 \cdot - 1 + 1 (x + 2)$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + 2 x = x^{2} + x \cdot - 1 - 3 x - 3 \cdot - 1 + 1 (x + 2)$

Step 3.3.1.3.1.2

Move $- 1$ to the left of $x$ .

$x^{2} + 2 x = x^{2} - 1 \cdot x - 3 x - 3 \cdot - 1 + 1 (x + 2)$

Step 3.3.1.3.1.3

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

$x^{2} + 2 x = x^{2} - x - 3 x - 3 \cdot - 1 + 1 (x + 2)$

Step 3.3.1.3.1.4

Multiply $- 3$ by $- 1$ .

$x^{2} + 2 x = x^{2} - x - 3 x + 3 + 1 (x + 2)$

Step 3.3.1.3.2

Subtract $3 x$ from $- x$ .

$x^{2} + 2 x = x^{2} - 4 x + 3 + 1 (x + 2)$

Step 3.3.1.4

Multiply $x + 2$ by $1$ .

$x^{2} + 2 x = x^{2} - 4 x + 3 + x + 2$

Step 3.3.2

Simplify by adding terms.

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Step 3.3.2.1

Add $- 4 x$ and $x$ .

$x^{2} + 2 x = x^{2} - 3 x + 3 + 2$

Step 3.3.2.2

Add $3$ and $2$ .

$x^{2} + 2 x = x^{2} - 3 x + 5$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

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

$x^{2} + 2 x - x^{2} = - 3 x + 5$

Step 4.1.2

Add $3 x$ to both sides of the equation.

$x^{2} + 2 x - x^{2} + 3 x = 5$

Step 4.1.3

Combine the opposite terms in $x^{2} + 2 x - x^{2} + 3 x$ .

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Step 4.1.3.1

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

$2 x + 0 + 3 x = 5$

Step 4.1.3.2

Add $2 x$ and $0$ .

$2 x + 3 x = 5$

Step 4.1.4

Add $2 x$ and $3 x$ .

$5 x = 5$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $5$ by $5$ .

$x = 1$