Algebra Examples

$\frac{5}{4 x^{2} - 8 x} + \frac{5 x}{x - 1}$

Step 1

Factor $4 x$ out of $4 x^{2} - 8 x$ .

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Step 1.1

Factor $4 x$ out of $4 x^{2}$ .

$\frac{5}{4 x (x) - 8 x} + \frac{5 x}{x - 1}$

Step 1.2

Factor $4 x$ out of $- 8 x$ .

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

Step 1.3

Factor $4 x$ out of $4 x (x) + 4 x (- 2)$ .

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

Step 2

To write $\frac{5}{4 x (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 3

To write $\frac{5 x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{4 x (x - 2)}{4 x (x - 2)}$ .

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

Step 4

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

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Step 4.1

Multiply $\frac{5}{4 x (x - 2)}$ by $\frac{x - 1}{x - 1}$ .

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

Step 4.2

Multiply $\frac{5 x}{x - 1}$ by $\frac{4 x (x - 2)}{4 x (x - 2)}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $5$ out of $5 (x - 1) + 5 x (4 x (x - 2))$ .

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Step 6.1.1

Factor $5$ out of $5 x (4 x (x - 2))$ .

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

Step 6.1.2

Factor $5$ out of $5 (x - 1) + 5 (x (4 x (x - 2)))$ .

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

Step 6.2

Rewrite using the commutative property of multiplication.

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

Step 6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1

Move $x$ .

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

Step 6.3.2

Multiply $x$ by $x$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.5.1

Move $x$ .

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

Step 6.5.2

Multiply $x$ by $x^{2}$ .

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Step 6.5.2.1

Raise $x$ to the power of $1$ .

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

Step 6.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5.3

Add $1$ and $2$ .

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

Step 6.6

Multiply $- 2$ by $4$ .

$\frac{5 (x - 1 + 4 x^{3} - 8 x^{2})}{4 x (x - 1) (x - 2)}$

Step 6.7

Reorder terms.

$\frac{5 (4 x^{3} - 8 x^{2} + x - 1)}{4 x (x - 1) (x - 2)}$