Solve 2x/4x^2-9a^2+x+3a/4x^2+6ax+x-3a/4x^2-6ax

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

Evaluate

$$\frac{1}{x}$$

Short Solution Steps

Factor 4x^{2}-9a^{2}. Factor 4x^{2}+6ax.

$$\frac{2x}{\left(2x-3a\right)\left(2x+3a\right)}+\frac{x+3a}{2x\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-3a\right)\left(2x+3a\right) and 2x\left(2x+3a\right) is 2x\left(2x-3a\right)\left(2x+3a\right). Multiply \frac{2x}{\left(2x-3a\right)\left(2x+3a\right)} times \frac{2x}{2x}. Multiply \frac{x+3a}{2x\left(2x+3a\right)} times \frac{2x-3a}{2x-3a}.

$$\frac{2x\times 2x}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{\left(x+3a\right)\left(2x-3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Since \frac{2x\times 2x}{2x\left(2x-3a\right)\left(2x+3a\right)} and \frac{\left(x+3a\right)\left(2x-3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x\times 2x+\left(x+3a\right)\left(2x-3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Do the multiplications in 2x\times 2x+\left(x+3a\right)\left(2x-3a\right).

$$\frac{4x^{2}+2x^{2}-3xa+6ax-9a^{2}}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Combine like terms in 4x^{2}+2x^{2}-3xa+6ax-9a^{2}.

$$\frac{6x^{2}-9a^{2}+3xa}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Factor the expressions that are not already factored in \frac{6x^{2}-9a^{2}+3xa}{2x\left(2x-3a\right)\left(2x+3a\right)}.

$$\frac{3\left(x-a\right)\left(2x+3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Cancel out 2x+3a in both numerator and denominator.

$$\frac{3\left(x-a\right)}{2x\left(2x-3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Factor 4x^{2}-6ax.

$$\frac{3\left(x-a\right)}{2x\left(2x-3a\right)}+\frac{x-3a}{2x\left(2x-3a\right)}$$

Since \frac{3\left(x-a\right)}{2x\left(2x-3a\right)} and \frac{x-3a}{2x\left(2x-3a\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(x-a\right)+x-3a}{2x\left(2x-3a\right)}$$

Do the multiplications in 3\left(x-a\right)+x-3a.

$$\frac{3x-3a+x-3a}{2x\left(2x-3a\right)}$$

Combine like terms in 3x-3a+x-3a.

$$\frac{4x-6a}{2x\left(2x-3a\right)}$$

Factor the expressions that are not already factored in \frac{4x-6a}{2x\left(2x-3a\right)}.

$$\frac{2\left(2x-3a\right)}{2x\left(2x-3a\right)}$$

Cancel out 2\left(2x-3a\right) in both numerator and denominator.

$$\frac{1}{x}$$

Expand

$$\frac{1}{x}$$

Short Solution Steps

Factor 4x^{2}-9a^{2}. Factor 4x^{2}+6ax.

$$\frac{2x}{\left(2x-3a\right)\left(2x+3a\right)}+\frac{x+3a}{2x\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-3a\right)\left(2x+3a\right) and 2x\left(2x+3a\right) is 2x\left(2x-3a\right)\left(2x+3a\right). Multiply \frac{2x}{\left(2x-3a\right)\left(2x+3a\right)} times \frac{2x}{2x}. Multiply \frac{x+3a}{2x\left(2x+3a\right)} times \frac{2x-3a}{2x-3a}.

$$\frac{2x\times 2x}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{\left(x+3a\right)\left(2x-3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Since \frac{2x\times 2x}{2x\left(2x-3a\right)\left(2x+3a\right)} and \frac{\left(x+3a\right)\left(2x-3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x\times 2x+\left(x+3a\right)\left(2x-3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Do the multiplications in 2x\times 2x+\left(x+3a\right)\left(2x-3a\right).

$$\frac{4x^{2}+2x^{2}-3xa+6ax-9a^{2}}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Combine like terms in 4x^{2}+2x^{2}-3xa+6ax-9a^{2}.

$$\frac{6x^{2}-9a^{2}+3xa}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Factor the expressions that are not already factored in \frac{6x^{2}-9a^{2}+3xa}{2x\left(2x-3a\right)\left(2x+3a\right)}.

$$\frac{3\left(x-a\right)\left(2x+3a\right)}{2x\left(2x-3a\right)\left(2x+3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Cancel out 2x+3a in both numerator and denominator.

$$\frac{3\left(x-a\right)}{2x\left(2x-3a\right)}+\frac{x-3a}{4x^{2}-6ax}$$

Factor 4x^{2}-6ax.

$$\frac{3\left(x-a\right)}{2x\left(2x-3a\right)}+\frac{x-3a}{2x\left(2x-3a\right)}$$

Since \frac{3\left(x-a\right)}{2x\left(2x-3a\right)} and \frac{x-3a}{2x\left(2x-3a\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(x-a\right)+x-3a}{2x\left(2x-3a\right)}$$

Do the multiplications in 3\left(x-a\right)+x-3a.

$$\frac{3x-3a+x-3a}{2x\left(2x-3a\right)}$$

Combine like terms in 3x-3a+x-3a.

$$\frac{4x-6a}{2x\left(2x-3a\right)}$$

Factor the expressions that are not already factored in \frac{4x-6a}{2x\left(2x-3a\right)}.

$$\frac{2\left(2x-3a\right)}{2x\left(2x-3a\right)}$$

Cancel out 2\left(2x-3a\right) in both numerator and denominator.

$$\frac{1}{x}$$