Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$- 4, 1$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Cancel the common factor of $x + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 2

To write $\frac{1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x - 4) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{x - 4}$ by $\frac{x - 2}{x - 2}$ .

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 4) (x - 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x - 3) (x - 4)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

Move $- 4$ to the left of $x$ .

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

Step 6.2.1.3

Multiply $- 3$ by $- 4$ .

$\frac{x - 2 + x^{2} - 4 x - 3 x + 12}{(x - 2) (x - 4)}$

Step 6.2.2

Subtract $3 x$ from $- 4 x$ .

$\frac{x - 2 + x^{2} - 7 x + 12}{(x - 2) (x - 4)}$

Step 6.3

Subtract $7 x$ from $x$ .

$\frac{- 6 x - 2 + x^{2} + 12}{(x - 2) (x - 4)}$

Step 6.4

Add $- 2$ and $12$ .

$\frac{- 6 x + x^{2} + 10}{(x - 2) (x - 4)}$

Step 6.5

Reorder terms.

$\frac{x^{2} - 6 x + 10}{(x - 2) (x - 4)}$