Algebra Examples

$\frac{5 x - 1}{x^{2} - 3 x + 2} + \frac{3}{2 x - 4}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 3 x + 2$ 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 $- 3$ .

$- 2, - 1$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Factor $2$ out of $2 x - 4$ .

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Step 1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.2.2

Factor $2$ out of $- 4$ .

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

Step 1.2.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $5$ .

$\frac{10 x - 1 \cdot 2 + 3 (x - 1)}{2 (x - 1) (x - 2)}$

Step 6.3

Multiply $- 1$ by $2$ .

$\frac{10 x - 2 + 3 (x - 1)}{2 (x - 1) (x - 2)}$

Step 6.4

Apply the distributive property.

$\frac{10 x - 2 + 3 x + 3 \cdot - 1}{2 (x - 1) (x - 2)}$

Step 6.5

Multiply $3$ by $- 1$ .

$\frac{10 x - 2 + 3 x - 3}{2 (x - 1) (x - 2)}$

Step 6.6

Add $10 x$ and $3 x$ .

$\frac{13 x - 2 - 3}{2 (x - 1) (x - 2)}$

Step 6.7

Subtract $3$ from $- 2$ .

$\frac{13 x - 5}{2 (x - 1) (x - 2)}$