Simplify (9x)/(6x^2)+(3x-4)/(2x)+(2x^2-2x)/(4x^2)

$\frac{9 x}{6 x^{2}} + \frac{3 x - 4}{2 x} + \frac{2 x^{2} - 2 x}{4 x^{2}}$

Step 1

Find the common denominator.

Multiply $\frac{9 x}{6 x^{2}}$ by $\frac{2}{2}$ .

$\frac{9 x}{6 x^{2}} \cdot \frac{2}{2} + \frac{3 x - 4}{2 x} + \frac{2 x^{2} - 2 x}{4 x^{2}}$

Multiply $\frac{9 x}{6 x^{2}}$ by $\frac{2}{2}$ .

$\frac{9 x \cdot 2}{6 x^{2} \cdot 2} + \frac{3 x - 4}{2 x} + \frac{2 x^{2} - 2 x}{4 x^{2}}$

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

$\frac{9 x \cdot 2}{6 x^{2} \cdot 2} + \frac{3 x - 4}{2 x} \cdot \frac{6 x}{6 x} + \frac{2 x^{2} - 2 x}{4 x^{2}}$

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

$\frac{9 x \cdot 2}{6 x^{2} \cdot 2} + \frac{(3 x - 4) (6 x)}{2 x (6 x)} + \frac{2 x^{2} - 2 x}{4 x^{2}}$

Multiply $\frac{2 x^{2} - 2 x}{4 x^{2}}$ by $\frac{3}{3}$ .

$\frac{9 x \cdot 2}{6 x^{2} \cdot 2} + \frac{(3 x - 4) (6 x)}{2 x (6 x)} + \frac{2 x^{2} - 2 x}{4 x^{2}} \cdot \frac{3}{3}$

Multiply $\frac{2 x^{2} - 2 x}{4 x^{2}}$ by $\frac{3}{3}$ .

$\frac{9 x \cdot 2}{6 x^{2} \cdot 2} + \frac{(3 x - 4) (6 x)}{2 x (6 x)} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

Reorder the factors of $6 x^{2} \cdot 2$ .

$\frac{9 x \cdot 2}{2 \cdot 6 x^{2}} + \frac{(3 x - 4) (6 x)}{2 x (6 x)} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

Multiply $2$ by $6$ .

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{2 x (6 x)} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

Reorder the factors of $2 x (6 x)$ .

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{2 \cdot 6 x \cdot x} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

Multiply $2$ by $6$ .

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 x \cdot x} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

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

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 (x^{1} x)} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

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

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 (x^{1} x^{1})} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

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

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 x^{1 + 1}} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

Add $1$ and $1$ .

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 x^{2}} + \frac{(2 x^{2} - 2 x) \cdot 3}{4 x^{2} \cdot 3}$

Reorder the factors of $4 x^{2} \cdot 3$ .

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 x^{2}} + \frac{(2 x^{2} - 2 x) \cdot 3}{3 \cdot 4 x^{2}}$

Multiply $3$ by $4$ .

$\frac{9 x \cdot 2}{12 x^{2}} + \frac{(3 x - 4) (6 x)}{12 x^{2}} + \frac{(2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Step 2

Combine the numerators over the common denominator.

$\frac{9 x \cdot 2 + (3 x - 4) (6 x) + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Step 3

Simplify each term.

Multiply $2$ by $9$ .

$\frac{18 x + (3 x - 4) (6 x) + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Apply the distributive property.

$\frac{18 x + 3 x (6 x) - 4 (6 x) + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{18 x + 3 \cdot 6 x \cdot x - 4 (6 x) + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Multiply $6$ by $- 4$ .

$\frac{18 x + 3 \cdot 6 x \cdot x - 24 x + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Simplify each term.

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

Move $x$ .

$\frac{18 x + 3 \cdot 6 (x \cdot x) - 24 x + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Multiply $x$ by $x$ .

$\frac{18 x + 3 \cdot 6 x^{2} - 24 x + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Multiply $3$ by $6$ .

$\frac{18 x + 18 x^{2} - 24 x + (2 x^{2} - 2 x) \cdot 3}{12 x^{2}}$

Apply the distributive property.

$\frac{18 x + 18 x^{2} - 24 x + 2 x^{2} \cdot 3 - 2 x \cdot 3}{12 x^{2}}$

Multiply $3$ by $2$ .

$\frac{18 x + 18 x^{2} - 24 x + 6 x^{2} - 2 x \cdot 3}{12 x^{2}}$

Multiply $3$ by $- 2$ .

$\frac{18 x + 18 x^{2} - 24 x + 6 x^{2} - 6 x}{12 x^{2}}$

Step 4

Simplify terms.

Subtract $24 x$ from $18 x$ .

$\frac{18 x^{2} - 6 x + 6 x^{2} - 6 x}{12 x^{2}}$

Add $18 x^{2}$ and $6 x^{2}$ .

$\frac{24 x^{2} - 6 x - 6 x}{12 x^{2}}$

Subtract $6 x$ from $- 6 x$ .

$\frac{24 x^{2} - 12 x}{12 x^{2}}$

Factor $12 x$ out of $24 x^{2} - 12 x$ .

Factor $12 x$ out of $24 x^{2}$ .

$\frac{12 x (2 x) - 12 x}{12 x^{2}}$

Factor $12 x$ out of $- 12 x$ .

$\frac{12 x (2 x) + 12 x (- 1)}{12 x^{2}}$

Factor $12 x$ out of $12 x (2 x) + 12 x (- 1)$ .

$\frac{12 x (2 x - 1)}{12 x^{2}}$

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 x (2 x - 1)}{12 x^{2}}$

Rewrite the expression.

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

Step 5

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (9x)/(6x^2)+(3x-4)/(2x)+(2x^2-2x)/(4x^2)

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