Precalculus Examples

$\frac{x}{x^{2} - 7 x + 6} - \frac{x}{x^{2} - 2 x - 24}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 7 x + 6$ 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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{x}{(x - 6) (x - 1)} - \frac{x}{x^{2} - 2 x - 24}$

Step 1.2

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x - 6) (x - 1) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 6) (x - 1) (x + 4)$ .

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

Step 4.4

Reorder the factors of $(x - 6) (x + 4) (x - 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $x (x + 4) - x (x - 1)$ .

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Step 6.1.1

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

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

Step 6.1.2

Factor $x$ out of $x (x + 4) + x (- (x - 1))$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- 1$ by $- 1$ .

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

Step 6.4

Subtract $x$ from $x$ .

$\frac{x (0 + 4 + 1)}{(x + 4) (x - 1) (x - 6)}$

Step 6.5

Add $0$ and $4$ .

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

Step 6.6

Add $4$ and $1$ .

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

Step 7

Move $5$ to the left of $x$ .

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