Algebra Examples

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

Step 1

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

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

$1, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $- \frac{x + 1}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Apply the distributive property.

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

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Move $x$ .

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

Step 5.4.1.1.2

Multiply $x$ by $x$ .

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

Step 5.4.1.2

Multiply $- 1$ by $1$ .

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

Step 5.4.1.3

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

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

Step 5.4.1.4

Multiply $- 1$ by $1$ .

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

Step 5.4.2

Subtract $x$ from $- x$ .

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

Step 5.5

Subtract $2 x$ from $x$ .

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

Step 5.6

Subtract $1$ from $2$ .

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

Step 5.7

Reorder terms.

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- x^{2}$ .

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

Step 6.2

Factor $- 1$ out of $- x$ .

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

Step 6.3

Factor $- 1$ out of $- (x^{2}) - (x)$ .

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

Step 6.4

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.5

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 1)$ .

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

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (x^{2} + x - 1)$ as $- 1 (x^{2} + x - 1)$ .

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

Step 6.6.2

Move the negative in front of the fraction.

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