Solve x/3x+1-2/3x

$$\frac { x } { 3 x + 1 } - \frac { 2 } { 3 x }$$

Evaluate

$$\frac{3x^{2}-6x-2}{3x\left(3x+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x+1 and 3x is 3x\left(3x+1\right). Multiply \frac{x}{3x+1} times \frac{3x}{3x}. Multiply \frac{2}{3x} times \frac{3x+1}{3x+1}.

$$\frac{x\times 3x}{3x\left(3x+1\right)}-\frac{2\left(3x+1\right)}{3x\left(3x+1\right)}$$

Since \frac{x\times 3x}{3x\left(3x+1\right)} and \frac{2\left(3x+1\right)}{3x\left(3x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 3x-2\left(3x+1\right)}{3x\left(3x+1\right)}$$

Do the multiplications in x\times 3x-2\left(3x+1\right).

$$\frac{3x^{2}-6x-2}{3x\left(3x+1\right)}$$

Factor the expressions that are not already factored in \frac{3x^{2}-6x-2}{3x\left(3x+1\right)}.

$$\frac{3\left(x-\left(-\frac{1}{3}\sqrt{15}+1\right)\right)\left(x-\left(\frac{1}{3}\sqrt{15}+1\right)\right)}{3x\left(3x+1\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{15}+1\right)\right)\left(x-\left(\frac{1}{3}\sqrt{15}+1\right)\right)}{x\left(3x+1\right)}$$

Expand x\left(3x+1\right).

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{15}+1\right)\right)\left(x-\left(\frac{1}{3}\sqrt{15}+1\right)\right)}{3x^{2}+x}$$

To find the opposite of -\frac{1}{3}\sqrt{15}+1, find the opposite of each term.

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{15}\right)-1\right)\left(x-\left(\frac{1}{3}\sqrt{15}+1\right)\right)}{3x^{2}+x}$$

The opposite of -\frac{1}{3}\sqrt{15} is \frac{1}{3}\sqrt{15}.

$$\frac{\left(x+\frac{1}{3}\sqrt{15}-1\right)\left(x-\left(\frac{1}{3}\sqrt{15}+1\right)\right)}{3x^{2}+x}$$

To find the opposite of \frac{1}{3}\sqrt{15}+1, find the opposite of each term.

$$\frac{\left(x+\frac{1}{3}\sqrt{15}-1\right)\left(x-\frac{1}{3}\sqrt{15}-1\right)}{3x^{2}+x}$$

Apply the distributive property by multiplying each term of x+\frac{1}{3}\sqrt{15}-1 by each term of x-\frac{1}{3}\sqrt{15}-1.

$$\frac{x^{2}+x\left(-\frac{1}{3}\right)\sqrt{15}-x+\frac{1}{3}\sqrt{15}x+\frac{1}{3}\sqrt{15}\left(-\frac{1}{3}\right)\sqrt{15}+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$\frac{x^{2}+x\left(-\frac{1}{3}\right)\sqrt{15}-x+\frac{1}{3}\sqrt{15}x+\frac{1}{3}\times 15\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Combine x\left(-\frac{1}{3}\right)\sqrt{15} and \frac{1}{3}\sqrt{15}x to get 0.

$$\frac{x^{2}-x+\frac{1}{3}\times 15\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Multiply \frac{1}{3} and 15 to get \frac{15}{3}.

$$\frac{x^{2}-x+\frac{15}{3}\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Divide 15 by 3 to get 5.

$$\frac{x^{2}-x+5\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Express 5\left(-\frac{1}{3}\right) as a single fraction.

$$\frac{x^{2}-x+\frac{5\left(-1\right)}{3}+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Multiply 5 and -1 to get -5.

$$\frac{x^{2}-x+\frac{-5}{3}+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Fraction \frac{-5}{3} can be rewritten as -\frac{5}{3} by extracting the negative sign.

$$\frac{x^{2}-x-\frac{5}{3}+\frac{1}{3}\sqrt{15}\left(-1\right)-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

$$\frac{x^{2}-x-\frac{5}{3}-\frac{1}{3}\sqrt{15}-x-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Combine -x and -x to get -2x.

$$\frac{x^{2}-2x-\frac{5}{3}-\frac{1}{3}\sqrt{15}-\left(-\frac{1}{3}\sqrt{15}\right)+1}{3x^{2}+x}$$

Multiply -1 and -\frac{1}{3} to get \frac{1}{3}.

$$\frac{x^{2}-2x-\frac{5}{3}-\frac{1}{3}\sqrt{15}+\frac{1}{3}\sqrt{15}+1}{3x^{2}+x}$$

Combine -\frac{1}{3}\sqrt{15} and \frac{1}{3}\sqrt{15} to get 0.

$$\frac{x^{2}-2x-\frac{5}{3}+1}{3x^{2}+x}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{x^{2}-2x-\frac{5}{3}+\frac{3}{3}}{3x^{2}+x}$$

Since -\frac{5}{3} and \frac{3}{3} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-2x+\frac{-5+3}{3}}{3x^{2}+x}$$

Add -5 and 3 to get -2.

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

Factor

$$\frac{\left(x-\left(-\frac{\sqrt{15}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{15}}{3}+1\right)\right)}{x\left(3x+1\right)}$$