Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2.3

Multiply $- 1$ by $- 2$ .

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

Step 1.3

Simplify terms.

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Step 1.3.1

Subtract $3 x$ from $7 x$ .

$\frac{4 x + 4 + 2}{x^{2} + 3 x + 2}$

Step 1.3.2

Add $4$ and $2$ .

$\frac{4 x + 6}{x^{2} + 3 x + 2}$

Step 1.3.3

Factor $2$ out of $4 x + 6$ .

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Step 1.3.3.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 6}{x^{2} + 3 x + 2}$

Step 1.3.3.2

Factor $2$ out of $6$ .

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

Step 1.3.3.3

Factor $2$ out of $2 (2 x) + 2 (3)$ .

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

Step 2

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

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Step 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 $2$ and whose sum is $3$ .

$1, 2$

Step 2.2

Write the factored form using these integers.

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