Algebra Examples

$\frac{x}{2 x - 1} + \frac{x + 2}{x}$

Step 1

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

$\frac{x}{2 x - 1} \cdot \frac{x}{x} + \frac{x + 2}{x}$

Step 2

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

$\frac{x}{2 x - 1} \cdot \frac{x}{x} + \frac{x + 2}{x} \cdot \frac{2 x - 1}{2 x - 1}$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Multiply $x$ by $x$ .

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

Step 5.2

Expand $(x + 2) (2 x - 1)$ using the FOIL Method.

Tap for more steps...

Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.2

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

Tap for more steps...

Step 5.3.1.2.1

Move $x$ .

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

Step 5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 5.3.1.3

Move $- 1$ to the left of $x$ .

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

Step 5.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.1.5

Multiply $2$ by $2$ .

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

Step 5.3.1.6

Multiply $2$ by $- 1$ .

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

Step 5.3.2

Add $- x$ and $4 x$ .

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

Step 5.4

Add $x^{2}$ and $2 x^{2}$ .

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