Algebra Examples

$\frac{x + 2}{x^{2} - 9} + \frac{x - 3}{x^{2} - 2 x - 3} = \frac{2 x - 3}{x^{2} + 4 x + 3}$

Step 1

Subtract $\frac{2 x - 3}{x^{2} + 4 x + 3}$ from both sides of the equation.

$\frac{x + 2}{x^{2} - 9} + \frac{x - 3}{x^{2} - 2 x - 3} - \frac{2 x - 3}{x^{2} + 4 x + 3} = 0$

Step 2

Simplify $\frac{x + 2}{x^{2} - 9} + \frac{x - 3}{x^{2} - 2 x - 3} - \frac{2 x - 3}{x^{2} + 4 x + 3}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the denominator.

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Step 2.1.1.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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 $- 2$ .

$- 3, 1$

Step 2.1.2.2

Write the factored form using these integers.

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

Step 2.1.3

Cancel the common factor of $x - 3$ .

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Step 2.1.3.1

Cancel the common factor.

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

Step 2.1.3.2

Rewrite the expression.

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

Step 2.1.4

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

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Step 2.1.4.1

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 $4$ .

$1, 3$

Step 2.1.4.2

Write the factored form using these integers.

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

Step 2.2

Find the common denominator.

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Step 2.2.1

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

$\frac{x + 2}{(x + 3) (x - 3)} \cdot \frac{x + 1}{x + 1} + \frac{1}{x + 1} - \frac{2 x - 3}{(x + 1) (x + 3)} = 0$

Step 2.2.2

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

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

Step 2.2.3

Multiply $\frac{1}{x + 1}$ by $\frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$ .

$\frac{(x + 2) (x + 1)}{(x + 3) (x - 3) (x + 1)} + \frac{1}{x + 1} \cdot \frac{(x + 3) (x - 3)}{(x + 3) (x - 3)} - \frac{2 x - 3}{(x + 1) (x + 3)} = 0$

Step 2.2.4

Multiply $\frac{1}{x + 1}$ by $\frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$ .

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

Step 2.2.5

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

$\frac{(x + 2) (x + 1)}{(x + 3) (x - 3) (x + 1)} + \frac{(x + 3) (x - 3)}{(x + 1) ((x + 3) (x - 3))} - (\frac{2 x - 3}{(x + 1) (x + 3)} \cdot \frac{x - 3}{x - 3}) = 0$

Step 2.2.6

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

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

Step 2.2.7

Reorder the factors of $(x + 1) ((x + 3) (x - 3))$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Expand $(x + 2) (x + 1)$ using the FOIL Method.

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Step 2.4.1.1

Apply the distributive property.

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

Step 2.4.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 + 2 (x + 1) + (x + 3) (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 + 2 x + 2 \cdot 1 + (x + 3) (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.2

Simplify and combine like terms.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 1 + 2 x + 2 \cdot 1 + (x + 3) (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.2.1.2

Multiply $x$ by $1$ .

$\frac{x^{2} + x + 2 x + 2 \cdot 1 + (x + 3) (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.2.1.3

Multiply $2$ by $1$ .

$\frac{x^{2} + x + 2 x + 2 + (x + 3) (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.2.2

Add $x$ and $2 x$ .

$\frac{x^{2} + 3 x + 2 + (x + 3) (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.3

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

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Step 2.4.3.1

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x (x - 3) + 3 (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.3.2

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x \cdot x + x \cdot - 3 + 3 (x - 3) - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.3.3

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3 - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.4

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 2.4.4.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$\frac{x^{2} + 3 x + 2 + x \cdot x - 3 x + 3 x + 3 \cdot - 3 - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.4.2

Add $- 3 x$ and $3 x$ .

$\frac{x^{2} + 3 x + 2 + x \cdot x + 0 + 3 \cdot - 3 - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.4.3

Add $x \cdot x$ and $0$ .

$\frac{x^{2} + 3 x + 2 + x \cdot x + 3 \cdot - 3 - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.5

Simplify each term.

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Step 2.4.5.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 x + 2 + x^{2} + 3 \cdot - 3 - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.5.2

Multiply $3$ by $- 3$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - (2 x - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.6

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 + (- (2 x) - - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.7

Multiply $2$ by $- 1$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 + (- 2 x - - 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.8

Multiply $- 1$ by $- 3$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 + (- 2 x + 3) (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.9

Expand $(- 2 x + 3) (x - 3)$ using the FOIL Method.

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Step 2.4.9.1

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x (x - 3) + 3 (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.9.2

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x \cdot x - 2 x \cdot - 3 + 3 (x - 3)}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.9.3

Apply the distributive property.

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x \cdot x - 2 x \cdot - 3 + 3 x + 3 \cdot - 3}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.10

Simplify and combine like terms.

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Step 2.4.10.1

Simplify each term.

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Step 2.4.10.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.10.1.1.1

Move $x$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 (x \cdot x) - 2 x \cdot - 3 + 3 x + 3 \cdot - 3}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.10.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x^{2} - 2 x \cdot - 3 + 3 x + 3 \cdot - 3}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.10.1.2

Multiply $- 3$ by $- 2$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x^{2} + 6 x + 3 x + 3 \cdot - 3}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.10.1.3

Multiply $3$ by $- 3$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x^{2} + 6 x + 3 x - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.4.10.2

Add $6 x$ and $3 x$ .

$\frac{x^{2} + 3 x + 2 + x^{2} - 9 - 2 x^{2} + 9 x - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.5

Add $x^{2}$ and $x^{2}$ .

$\frac{2 x^{2} + 3 x + 2 - 9 - 2 x^{2} + 9 x - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.6

Combine the opposite terms in $2 x^{2} + 3 x + 2 - 9 - 2 x^{2} + 9 x - 9$ .

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Step 2.6.1

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

$\frac{3 x + 2 - 9 + 0 + 9 x - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.6.2

Add $3 x + 2 - 9$ and $0$ .

$\frac{3 x + 2 - 9 + 9 x - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.7

Add $3 x$ and $9 x$ .

$\frac{12 x + 2 - 9 - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.8

Subtract $9$ from $2$ .

$\frac{12 x - 7 - 9}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.9

Subtract $9$ from $- 7$ .

$\frac{12 x - 16}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.10

Factor $4$ out of $12 x - 16$ .

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Step 2.10.1

Factor $4$ out of $12 x$ .

$\frac{4 (3 x) - 16}{(x + 1) (x + 3) (x - 3)} = 0$

Step 2.10.2

Factor $4$ out of $- 16$ .

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

Step 2.10.3

Factor $4$ out of $4 (3 x) + 4 (- 4)$ .

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

Step 3

Set the numerator equal to zero.

$4 (3 x - 4) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $4 (3 x - 4) = 0$ by $4$ and simplify.

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Step 4.1.1

Divide each term in $4 (3 x - 4) = 0$ by $4$ .

$\frac{4 (3 x - 4)}{4} = \frac{0}{4}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{4 (3 x - 4)}{4} = \frac{0}{4}$

Step 4.1.2.1.2

Divide $3 x - 4$ by $1$ .

$3 x - 4 = \frac{0}{4}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $4$ .

$3 x - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$3 x = 4$

Step 4.3

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 4.3.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$