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

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

Step 1

Multiply the numerator and denominator of the complex fraction by $18$ .

Multiply $\frac{\frac{x^{2}}{3} - \frac{2}{9}}{\frac{x + 3}{2} + \frac{2 x}{3}}$ by $\frac{18}{18}$ .

$\frac{18}{18} \cdot \frac{\frac{x^{2}}{3} - \frac{2}{9}}{\frac{x + 3}{2} + \frac{2 x}{3}}$

Combine.

$\frac{18 (\frac{x^{2}}{3} - \frac{2}{9})}{18 (\frac{x + 3}{2} + \frac{2 x}{3})}$

Step 2

Apply the distributive property.

$\frac{18 \frac{x^{2}}{3} + 18 (- \frac{2}{9})}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $3$ .

Factor $3$ out of $18$ .

$\frac{3 (6) \frac{x^{2}}{3} + 18 (- \frac{2}{9})}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Cancel the common factor.

$\frac{3 \cdot 6 \frac{x^{2}}{3} + 18 (- \frac{2}{9})}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Rewrite the expression.

$\frac{6 x^{2} + 18 (- \frac{2}{9})}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{2}{9}$ into the numerator.

$\frac{6 x^{2} + 18 (\frac{- 2}{9})}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Factor $9$ out of $18$ .

$\frac{6 x^{2} + 9 (2) \frac{- 2}{9}}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Cancel the common factor.

$\frac{6 x^{2} + 9 \cdot 2 \frac{- 2}{9}}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Rewrite the expression.

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

Multiply $2$ by $- 2$ .

$\frac{6 x^{2} - 4}{18 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Cancel the common factor of $2$ .

Factor $2$ out of $18$ .

$\frac{6 x^{2} - 4}{2 (9) \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Cancel the common factor.

$\frac{6 x^{2} - 4}{2 \cdot 9 \frac{x + 3}{2} + 18 \frac{2 x}{3}}$

Rewrite the expression.

$\frac{6 x^{2} - 4}{9 (x + 3) + 18 \frac{2 x}{3}}$

Cancel the common factor of $3$ .

Factor $3$ out of $18$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $6$ .

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

Step 4

Factor $2$ out of $6 x^{2} - 4$ .

Factor $2$ out of $6 x^{2}$ .

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

Factor $2$ out of $- 4$ .

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

Factor $2$ out of $2 (3 x^{2}) + 2 (- 2)$ .

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

Step 5

Simplify the denominator.

Factor $3$ out of $9 (x + 3) + 12 x$ .

Factor $3$ out of $9 (x + 3)$ .

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

Factor $3$ out of $12 x$ .

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

Factor $3$ out of $3 (3 (x + 3)) + 3 (4 x)$ .

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

Apply the distributive property.

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

Multiply $3$ by $3$ .

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

Add $3 x$ and $4 x$ .

$\frac{2 (3 x^{2} - 2)}{3 (7 x + 9)}$

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

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