Precalculus Examples

$\frac{x}{x^{2} + x - 2} - \frac{6}{x^{2} - 5 x + 4}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{x}{(x - 1) (x + 2)} - \frac{6}{x^{2} - 5 x + 4}$

Step 1.2

Factor $x^{2} - 5 x + 4$ using the AC method.

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Step 1.2.1

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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{x}{(x - 1) (x + 2)} - \frac{6}{(x - 4) (x - 1)}$

Step 2

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

$\frac{x}{(x - 1) (x + 2)} \cdot \frac{x - 4}{x - 4} - \frac{6}{(x - 4) (x - 1)}$

Step 3

To write $- \frac{6}{(x - 4) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{x}{(x - 1) (x + 2)} \cdot \frac{x - 4}{x - 4} - \frac{6}{(x - 4) (x - 1)} \cdot \frac{x + 2}{x + 2}$

Step 4

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

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Step 4.1

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

$\frac{x (x - 4)}{(x - 1) (x + 2) (x - 4)} - \frac{6}{(x - 4) (x - 1)} \cdot \frac{x + 2}{x + 2}$

Step 4.2

Multiply $\frac{6}{(x - 4) (x - 1)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{x (x - 4)}{(x - 1) (x + 2) (x - 4)} - \frac{6 (x + 2)}{(x - 4) (x - 1) (x + 2)}$

Step 4.3

Reorder the factors of $(x - 1) (x + 2) (x - 4)$ .

$\frac{x (x - 4)}{(x + 2) (x - 1) (x - 4)} - \frac{6 (x + 2)}{(x - 4) (x - 1) (x + 2)}$

Step 4.4

Reorder the factors of $(x - 4) (x - 1) (x + 2)$ .

$\frac{x (x - 4)}{(x + 2) (x - 1) (x - 4)} - \frac{6 (x + 2)}{(x + 2) (x - 1) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (x - 4) - 6 (x + 2)}{(x + 2) (x - 1) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 - 6 (x + 2)}{(x + 2) (x - 1) (x - 4)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 4 - 6 (x + 2)}{(x + 2) (x - 1) (x - 4)}$

Step 6.3

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

$\frac{x^{2} - 4 \cdot x - 6 (x + 2)}{(x + 2) (x - 1) (x - 4)}$

Step 6.4

Apply the distributive property.

$\frac{x^{2} - 4 x - 6 x - 6 \cdot 2}{(x + 2) (x - 1) (x - 4)}$

Step 6.5

Multiply $- 6$ by $2$ .

$\frac{x^{2} - 4 x - 6 x - 12}{(x + 2) (x - 1) (x - 4)}$

Step 6.6

Subtract $6 x$ from $- 4 x$ .

$\frac{x^{2} - 10 x - 12}{(x + 2) (x - 1) (x - 4)}$