Algebra Examples

$\frac{2}{x + 3} + \frac{3}{x + 4} = \frac{7}{x^{2} + 7 x + 12}$

Step 1

Subtract $\frac{7}{x^{2} + 7 x + 12}$ from both sides of the equation.

$\frac{2}{x + 3} + \frac{3}{x + 4} - \frac{7}{x^{2} + 7 x + 12} = 0$

Step 2

Simplify $\frac{2}{x + 3} + \frac{3}{x + 4} - \frac{7}{x^{2} + 7 x + 12}$ .

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Step 2.1

Factor $x^{2} + 7 x + 12$ 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 $12$ and whose sum is $7$ .

$3, 4$

Step 2.1.2

Write the factored form using these integers.

$\frac{2}{x + 3} + \frac{3}{x + 4} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.2

To write $\frac{2}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{2}{x + 3} \cdot \frac{x + 4}{x + 4} + \frac{3}{x + 4} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.3

To write $\frac{3}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{2}{x + 3} \cdot \frac{x + 4}{x + 4} + \frac{3}{x + 4} \cdot \frac{x + 3}{x + 3} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.4

Write each expression with a common denominator of $(x + 3) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{2}{x + 3}$ by $\frac{x + 4}{x + 4}$ .

$\frac{2 (x + 4)}{(x + 3) (x + 4)} + \frac{3}{x + 4} \cdot \frac{x + 3}{x + 3} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.4.2

Multiply $\frac{3}{x + 4}$ by $\frac{x + 3}{x + 3}$ .

$\frac{2 (x + 4)}{(x + 3) (x + 4)} + \frac{3 (x + 3)}{(x + 4) (x + 3)} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.4.3

Reorder the factors of $(x + 4) (x + 3)$ .

$\frac{2 (x + 4)}{(x + 3) (x + 4)} + \frac{3 (x + 3)}{(x + 3) (x + 4)} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{2 (x + 4) + 3 (x + 3)}{(x + 3) (x + 4)} - \frac{7}{(x + 3) (x + 4)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{2 (x + 4) + 3 (x + 3) - 7}{(x + 3) (x + 4)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{2 x + 2 \cdot 4 + 3 (x + 3) - 7}{(x + 3) (x + 4)} = 0$

Step 2.7.2

Multiply $2$ by $4$ .

$\frac{2 x + 8 + 3 (x + 3) - 7}{(x + 3) (x + 4)} = 0$

Step 2.7.3

Apply the distributive property.

$\frac{2 x + 8 + 3 x + 3 \cdot 3 - 7}{(x + 3) (x + 4)} = 0$

Step 2.7.4

Multiply $3$ by $3$ .

$\frac{2 x + 8 + 3 x + 9 - 7}{(x + 3) (x + 4)} = 0$

Step 2.8

Add $2 x$ and $3 x$ .

$\frac{5 x + 8 + 9 - 7}{(x + 3) (x + 4)} = 0$

Step 2.9

Add $8$ and $9$ .

$\frac{5 x + 17 - 7}{(x + 3) (x + 4)} = 0$

Step 2.10

Subtract $7$ from $17$ .

$\frac{5 x + 10}{(x + 3) (x + 4)} = 0$

Step 2.11

Factor $5$ out of $5 x + 10$ .

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Step 2.11.1

Factor $5$ out of $5 x$ .

$\frac{5 (x) + 10}{(x + 3) (x + 4)} = 0$

Step 2.11.2

Factor $5$ out of $10$ .

$\frac{5 x + 5 \cdot 2}{(x + 3) (x + 4)} = 0$

Step 2.11.3

Factor $5$ out of $5 x + 5 \cdot 2$ .

$\frac{5 (x + 2)}{(x + 3) (x + 4)} = 0$

Step 3

Set the numerator equal to zero.

$5 (x + 2) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $5 (x + 2) = 0$ by $5$ and simplify.

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Step 4.1.1

Divide each term in $5 (x + 2) = 0$ by $5$ .

$\frac{5 (x + 2)}{5} = \frac{0}{5}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{5 (x + 2)}{5} = \frac{0}{5}$

Step 4.1.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{0}{5}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $5$ .

$x + 2 = 0$

Step 4.2

Subtract $2$ from both sides of the equation.

$x = - 2$