Algebra Examples

$\frac{3}{x^{2} + 5 x + 6} + \frac{x - 1}{x + 2} = \frac{7}{x + 3}$

Step 1

Subtract $\frac{7}{x + 3}$ from both sides of the equation.

$\frac{3}{x^{2} + 5 x + 6} + \frac{x - 1}{x + 2} - \frac{7}{x + 3} = 0$

Step 2

Simplify $\frac{3}{x^{2} + 5 x + 6} + \frac{x - 1}{x + 2} - \frac{7}{x + 3}$ .

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Step 2.1

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

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

$2, 3$

Step 2.1.2

Write the factored form using these integers.

$\frac{3}{(x + 2) (x + 3)} + \frac{x - 1}{x + 2} - \frac{7}{x + 3} = 0$

Step 2.2

Find the common denominator.

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Step 2.2.1

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

$\frac{3}{(x + 2) (x + 3)} + \frac{x - 1}{x + 2} \cdot \frac{x + 3}{x + 3} - \frac{7}{x + 3} = 0$

Step 2.2.2

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

$\frac{3}{(x + 2) (x + 3)} + \frac{(x - 1) (x + 3)}{(x + 2) (x + 3)} - \frac{7}{x + 3} = 0$

Step 2.2.3

Multiply $\frac{7}{x + 3}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3}{(x + 2) (x + 3)} + \frac{(x - 1) (x + 3)}{(x + 2) (x + 3)} - (\frac{7}{x + 3} \cdot \frac{x + 2}{x + 2}) = 0$

Step 2.2.4

Multiply $\frac{7}{x + 3}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3}{(x + 2) (x + 3)} + \frac{(x - 1) (x + 3)}{(x + 2) (x + 3)} - \frac{7 (x + 2)}{(x + 3) (x + 2)} = 0$

Step 2.2.5

Reorder the factors of $(x + 3) (x + 2)$ .

$\frac{3}{(x + 2) (x + 3)} + \frac{(x - 1) (x + 3)}{(x + 2) (x + 3)} - \frac{7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{3 + (x - 1) (x + 3) - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

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

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Step 2.4.1.1

Apply the distributive property.

$\frac{3 + x (x + 3) - 1 (x + 3) - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.1.2

Apply the distributive property.

$\frac{3 + x \cdot x + x \cdot 3 - 1 (x + 3) - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.1.3

Apply the distributive property.

$\frac{3 + x \cdot x + x \cdot 3 - 1 x - 1 \cdot 3 - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

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 $x$ .

$\frac{3 + x^{2} + x \cdot 3 - 1 x - 1 \cdot 3 - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.2.1.2

Move $3$ to the left of $x$ .

$\frac{3 + x^{2} + 3 \cdot x - 1 x - 1 \cdot 3 - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.2.1.3

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

$\frac{3 + x^{2} + 3 x - x - 1 \cdot 3 - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.2.1.4

Multiply $- 1$ by $3$ .

$\frac{3 + x^{2} + 3 x - x - 3 - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.2.2

Subtract $x$ from $3 x$ .

$\frac{3 + x^{2} + 2 x - 3 - 7 (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.4.3

Apply the distributive property.

$\frac{3 + x^{2} + 2 x - 3 - 7 x - 7 \cdot 2}{(x + 2) (x + 3)} = 0$

Step 2.4.4

Multiply $- 7$ by $2$ .

$\frac{3 + x^{2} + 2 x - 3 - 7 x - 14}{(x + 2) (x + 3)} = 0$

Step 2.5

Combine the opposite terms in $3 + x^{2} + 2 x - 3 - 7 x - 14$ .

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Step 2.5.1

Subtract $3$ from $3$ .

$\frac{x^{2} + 2 x + 0 - 7 x - 14}{(x + 2) (x + 3)} = 0$

Step 2.5.2

Add $x^{2} + 2 x$ and $0$ .

$\frac{x^{2} + 2 x - 7 x - 14}{(x + 2) (x + 3)} = 0$

Step 2.6

Subtract $7 x$ from $2 x$ .

$\frac{x^{2} - 5 x - 14}{(x + 2) (x + 3)} = 0$

Step 2.7

Factor $x^{2} - 5 x - 14$ using the AC method.

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Step 2.7.1

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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 2.7.2

Write the factored form using these integers.

$\frac{(x - 7) (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.8

Cancel the common factor of $x + 2$ .

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Step 2.8.1

Cancel the common factor.

$\frac{(x - 7) (x + 2)}{(x + 2) (x + 3)} = 0$

Step 2.8.2

Rewrite the expression.

$\frac{x - 7}{x + 3} = 0$

Step 3

Set the numerator equal to zero.

$x - 7 = 0$

Step 4

Add $7$ to both sides of the equation.

$x = 7$