Pre-Algebra Examples

$\frac{6 x}{x^{2} - 5 x + 6} - \frac{24}{x^{2} - 6 x + 8}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 5 x + 6$ 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 $- 5$ .

$- 3, - 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{6 x}{(x - 3) (x - 2)} - \frac{24}{x^{2} - 6 x + 8}$

Step 1.2

Factor $x^{2} - 6 x + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{6 x}{(x - 3) (x - 2)} - \frac{24}{(x - 4) (x - 2)}$

Step 2

To write $\frac{6 x}{(x - 3) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{6 x}{(x - 3) (x - 2)} \cdot \frac{x - 4}{x - 4} - \frac{24}{(x - 4) (x - 2)}$

Step 3

To write $- \frac{24}{(x - 4) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{6 x}{(x - 3) (x - 2)} \cdot \frac{x - 4}{x - 4} - \frac{24}{(x - 4) (x - 2)} \cdot \frac{x - 3}{x - 3}$

Step 4

Write each expression with a common denominator of $(x - 3) (x - 2) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6 x}{(x - 3) (x - 2)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{6 x (x - 4)}{(x - 3) (x - 2) (x - 4)} - \frac{24}{(x - 4) (x - 2)} \cdot \frac{x - 3}{x - 3}$

Step 4.2

Multiply $\frac{24}{(x - 4) (x - 2)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{6 x (x - 4)}{(x - 3) (x - 2) (x - 4)} - \frac{24 (x - 3)}{(x - 4) (x - 2) (x - 3)}$

Step 4.3

Reorder the factors of $(x - 3) (x - 2) (x - 4)$ .

$\frac{6 x (x - 4)}{(x - 2) (x - 3) (x - 4)} - \frac{24 (x - 3)}{(x - 4) (x - 2) (x - 3)}$

Step 4.4

Reorder the factors of $(x - 4) (x - 2) (x - 3)$ .

$\frac{6 x (x - 4)}{(x - 2) (x - 3) (x - 4)} - \frac{24 (x - 3)}{(x - 2) (x - 3) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 x (x - 4) - 24 (x - 3)}{(x - 2) (x - 3) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $6$ out of $6 x (x - 4) - 24 (x - 3)$ .

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Step 6.1.1

Factor $6$ out of $6 x (x - 4)$ .

$\frac{6 (x (x - 4)) - 24 (x - 3)}{(x - 2) (x - 3) (x - 4)}$

Step 6.1.2

Factor $6$ out of $- 24 (x - 3)$ .

$\frac{6 (x (x - 4)) + 6 (- 4 (x - 3))}{(x - 2) (x - 3) (x - 4)}$

Step 6.1.3

Factor $6$ out of $6 (x (x - 4)) + 6 (- 4 (x - 3))$ .

$\frac{6 (x (x - 4) - 4 (x - 3))}{(x - 2) (x - 3) (x - 4)}$

Step 6.2

Apply the distributive property.

$\frac{6 (x \cdot x + x \cdot - 4 - 4 (x - 3))}{(x - 2) (x - 3) (x - 4)}$

Step 6.3

Multiply $x$ by $x$ .

$\frac{6 (x^{2} + x \cdot - 4 - 4 (x - 3))}{(x - 2) (x - 3) (x - 4)}$

Step 6.4

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

$\frac{6 (x^{2} - 4 \cdot x - 4 (x - 3))}{(x - 2) (x - 3) (x - 4)}$

Step 6.5

Apply the distributive property.

$\frac{6 (x^{2} - 4 x - 4 x - 4 \cdot - 3)}{(x - 2) (x - 3) (x - 4)}$

Step 6.6

Multiply $- 4$ by $- 3$ .

$\frac{6 (x^{2} - 4 x - 4 x + 12)}{(x - 2) (x - 3) (x - 4)}$

Step 6.7

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

$\frac{6 (x^{2} - 8 x + 12)}{(x - 2) (x - 3) (x - 4)}$

Step 6.8

Factor $x^{2} - 8 x + 12$ using the AC method.

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Step 6.8.1

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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 6.8.2

Write the factored form using these integers.

$\frac{6 ((x - 6) (x - 2))}{(x - 2) (x - 3) (x - 4)}$

$\frac{6 (x - 6) (x - 2)}{(x - 2) (x - 3) (x - 4)}$

Step 7

Cancel the common factor of $x - 2$ .

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Step 7.1

Cancel the common factor.

$\frac{6 (x - 6) (x - 2)}{(x - 2) (x - 3) (x - 4)}$

Step 7.2

Rewrite the expression.

$\frac{6 (x - 6)}{(x - 3) (x - 4)}$