Algebra Examples

$\frac{5}{2 x - 6} - \frac{x + 2}{x^{2} - 4 x + 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 x - 6$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{5}{2 (x) - 6} - \frac{x + 2}{x^{2} - 4 x + 3}$

Step 1.1.2

Factor $2$ out of $- 6$ .

$\frac{5}{2 x + 2 \cdot - 3} - \frac{x + 2}{x^{2} - 4 x + 3}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$\frac{5}{2 (x - 3)} - \frac{x + 2}{x^{2} - 4 x + 3}$

Step 1.2

Factor $x^{2} - 4 x + 3$ 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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{5}{2 (x - 3)} - \frac{x + 2}{(x - 3) (x - 1)}$

Step 2

To write $\frac{5}{2 (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{5}{2 (x - 3)} \cdot \frac{x - 1}{x - 1} - \frac{x + 2}{(x - 3) (x - 1)}$

Step 3

To write $- \frac{x + 2}{(x - 3) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2 (x - 3)} \cdot \frac{x - 1}{x - 1} - \frac{x + 2}{(x - 3) (x - 1)} \cdot \frac{2}{2}$

Step 4

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

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Step 4.1

Multiply $\frac{5}{2 (x - 3)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{5 (x - 1)}{2 (x - 3) (x - 1)} - \frac{x + 2}{(x - 3) (x - 1)} \cdot \frac{2}{2}$

Step 4.2

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

$\frac{5 (x - 1)}{2 (x - 3) (x - 1)} - \frac{(x + 2) \cdot 2}{(x - 3) (x - 1) \cdot 2}$

Step 4.3

Reorder the factors of $2 (x - 3) (x - 1)$ .

$\frac{5 (x - 1)}{2 (x - 1) (x - 3)} - \frac{(x + 2) \cdot 2}{(x - 3) (x - 1) \cdot 2}$

Step 4.4

Reorder the factors of $(x - 3) (x - 1) \cdot 2$ .

$\frac{5 (x - 1)}{2 (x - 1) (x - 3)} - \frac{(x + 2) \cdot 2}{2 (x - 1) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (x - 1) - (x + 2) \cdot 2}{2 (x - 1) (x - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{5 x + 5 \cdot - 1 - (x + 2) \cdot 2}{2 (x - 1) (x - 3)}$

Step 6.2

Multiply $5$ by $- 1$ .

$\frac{5 x - 5 - (x + 2) \cdot 2}{2 (x - 1) (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{5 x - 5 + (- x - 1 \cdot 2) \cdot 2}{2 (x - 1) (x - 3)}$

Step 6.4

Multiply $- 1$ by $2$ .

$\frac{5 x - 5 + (- x - 2) \cdot 2}{2 (x - 1) (x - 3)}$

Step 6.5

Apply the distributive property.

$\frac{5 x - 5 - x \cdot 2 - 2 \cdot 2}{2 (x - 1) (x - 3)}$

Step 6.6

Multiply $2$ by $- 1$ .

$\frac{5 x - 5 - 2 x - 2 \cdot 2}{2 (x - 1) (x - 3)}$

Step 6.7

Multiply $- 2$ by $2$ .

$\frac{5 x - 5 - 2 x - 4}{2 (x - 1) (x - 3)}$

Step 6.8

Subtract $2 x$ from $5 x$ .

$\frac{3 x - 5 - 4}{2 (x - 1) (x - 3)}$

Step 6.9

Subtract $4$ from $- 5$ .

$\frac{3 x - 9}{2 (x - 1) (x - 3)}$

Step 6.10

Factor $3$ out of $3 x - 9$ .

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Step 6.10.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) - 9}{2 (x - 1) (x - 3)}$

Step 6.10.2

Factor $3$ out of $- 9$ .

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

Step 6.10.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

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

Step 7

Cancel the common factor of $x - 3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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