Solve 14/x-3-5/x+2div9x/6

$$\frac { 14 } { x - 3 } - \frac { 5 } { x + 2 } \div \frac { 9 x } { 6 }$$

Evaluate

$$\frac{2\left(21x^{2}+37x+15\right)}{3x\left(x-3\right)\left(x+2\right)}$$

Short Solution Steps

Divide \frac{5}{x+2} by \frac{9x}{6} by multiplying \frac{5}{x+2} by the reciprocal of \frac{9x}{6}.

$$\frac{14}{x-3}-\frac{5\times 6}{\left(x+2\right)\times 9x}$$

Cancel out 3 in both numerator and denominator.

$$\frac{14}{x-3}-\frac{2\times 5}{3x\left(x+2\right)}$$

Multiply 2 and 5 to get 10.

$$\frac{14}{x-3}-\frac{10}{3x\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 3x\left(x+2\right) is 3x\left(x-3\right)\left(x+2\right). Multiply \frac{14}{x-3} times \frac{3x\left(x+2\right)}{3x\left(x+2\right)}. Multiply \frac{10}{3x\left(x+2\right)} times \frac{x-3}{x-3}.

$$\frac{14\times 3x\left(x+2\right)}{3x\left(x-3\right)\left(x+2\right)}-\frac{10\left(x-3\right)}{3x\left(x-3\right)\left(x+2\right)}$$

Since \frac{14\times 3x\left(x+2\right)}{3x\left(x-3\right)\left(x+2\right)} and \frac{10\left(x-3\right)}{3x\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{14\times 3x\left(x+2\right)-10\left(x-3\right)}{3x\left(x-3\right)\left(x+2\right)}$$

Do the multiplications in 14\times 3x\left(x+2\right)-10\left(x-3\right).

$$\frac{42x^{2}+84x-10x+30}{3x\left(x-3\right)\left(x+2\right)}$$

Combine like terms in 42x^{2}+84x-10x+30.

$$\frac{42x^{2}+74x+30}{3x\left(x-3\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{42x^{2}+74x+30}{3x\left(x-3\right)\left(x+2\right)}.

$$\frac{2\times 21\left(x-\left(-\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{3x\left(x-3\right)\left(x+2\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\times 7\left(x-\left(-\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{x\left(x-3\right)\left(x+2\right)}$$

Expand x\left(x-3\right)\left(x+2\right).

$$\frac{2\times 7\left(x-\left(-\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{x^{3}-x^{2}-6x}$$

Multiply 2 and 7 to get 14.

$$\frac{14\left(x-\left(-\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{x^{3}-x^{2}-6x}$$

To find the opposite of -\frac{1}{42}\sqrt{109}-\frac{37}{42}, find the opposite of each term.

$$\frac{14\left(x-\left(-\frac{1}{42}\sqrt{109}\right)-\left(-\frac{37}{42}\right)\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{x^{3}-x^{2}-6x}$$

The opposite of -\frac{1}{42}\sqrt{109} is \frac{1}{42}\sqrt{109}.

$$\frac{14\left(x+\frac{1}{42}\sqrt{109}-\left(-\frac{37}{42}\right)\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{x^{3}-x^{2}-6x}$$

The opposite of -\frac{37}{42} is \frac{37}{42}.

$$\frac{14\left(x+\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)\left(x-\left(\frac{1}{42}\sqrt{109}-\frac{37}{42}\right)\right)}{x^{3}-x^{2}-6x}$$

To find the opposite of \frac{1}{42}\sqrt{109}-\frac{37}{42}, find the opposite of each term.

$$\frac{14\left(x+\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)\left(x-\frac{1}{42}\sqrt{109}-\left(-\frac{37}{42}\right)\right)}{x^{3}-x^{2}-6x}$$

The opposite of -\frac{37}{42} is \frac{37}{42}.

$$\frac{14\left(x+\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Use the distributive property to multiply 14 by x+\frac{1}{42}\sqrt{109}+\frac{37}{42}.

$$\frac{\left(14x+14\times \frac{1}{42}\sqrt{109}+14\times \frac{37}{42}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Multiply 14 and \frac{1}{42} to get \frac{14}{42}.

$$\frac{\left(14x+\frac{14}{42}\sqrt{109}+14\times \frac{37}{42}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Reduce the fraction \frac{14}{42} to lowest terms by extracting and canceling out 14.

$$\frac{\left(14x+\frac{1}{3}\sqrt{109}+14\times \frac{37}{42}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Express 14\times \frac{37}{42} as a single fraction.

$$\frac{\left(14x+\frac{1}{3}\sqrt{109}+\frac{14\times 37}{42}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Multiply 14 and 37 to get 518.

$$\frac{\left(14x+\frac{1}{3}\sqrt{109}+\frac{518}{42}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Reduce the fraction \frac{518}{42} to lowest terms by extracting and canceling out 14.

$$\frac{\left(14x+\frac{1}{3}\sqrt{109}+\frac{37}{3}\right)\left(x-\frac{1}{42}\sqrt{109}+\frac{37}{42}\right)}{x^{3}-x^{2}-6x}$$

Apply the distributive property by multiplying each term of 14x+\frac{1}{3}\sqrt{109}+\frac{37}{3} by each term of x-\frac{1}{42}\sqrt{109}+\frac{37}{42}.

$$\frac{14x^{2}+14x\left(-\frac{1}{42}\right)\sqrt{109}+14x\times \frac{37}{42}+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\sqrt{109}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply \sqrt{109} and \sqrt{109} to get 109.

$$\frac{14x^{2}+14x\left(-\frac{1}{42}\right)\sqrt{109}+14x\times \frac{37}{42}+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Express 14\left(-\frac{1}{42}\right) as a single fraction.

$$\frac{14x^{2}+\frac{14\left(-1\right)}{42}x\sqrt{109}+14x\times \frac{37}{42}+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply 14 and -1 to get -14.

$$\frac{14x^{2}+\frac{-14}{42}x\sqrt{109}+14x\times \frac{37}{42}+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Reduce the fraction \frac{-14}{42} to lowest terms by extracting and canceling out 14.

$$\frac{14x^{2}-\frac{1}{3}x\sqrt{109}+14x\times \frac{37}{42}+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Express 14\times \frac{37}{42} as a single fraction.

$$\frac{14x^{2}-\frac{1}{3}x\sqrt{109}+\frac{14\times 37}{42}x+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply 14 and 37 to get 518.

$$\frac{14x^{2}-\frac{1}{3}x\sqrt{109}+\frac{518}{42}x+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Reduce the fraction \frac{518}{42} to lowest terms by extracting and canceling out 14.

$$\frac{14x^{2}-\frac{1}{3}x\sqrt{109}+\frac{37}{3}x+\frac{1}{3}\sqrt{109}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Combine -\frac{1}{3}x\sqrt{109} and \frac{1}{3}\sqrt{109}x to get 0.

$$\frac{14x^{2}+\frac{37}{3}x+\frac{1}{3}\times 109\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply \frac{1}{3} and 109 to get \frac{109}{3}.

$$\frac{14x^{2}+\frac{37}{3}x+\frac{109}{3}\left(-\frac{1}{42}\right)+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply \frac{109}{3} times -\frac{1}{42} by multiplying numerator times numerator and denominator times denominator.

$$\frac{14x^{2}+\frac{37}{3}x+\frac{109\left(-1\right)}{3\times 42}+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Do the multiplications in the fraction \frac{109\left(-1\right)}{3\times 42}.

$$\frac{14x^{2}+\frac{37}{3}x+\frac{-109}{126}+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Fraction \frac{-109}{126} can be rewritten as -\frac{109}{126} by extracting the negative sign.

$$\frac{14x^{2}+\frac{37}{3}x-\frac{109}{126}+\frac{1}{3}\sqrt{109}\times \frac{37}{42}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply \frac{1}{3} times \frac{37}{42} by multiplying numerator times numerator and denominator times denominator.

$$\frac{14x^{2}+\frac{37}{3}x-\frac{109}{126}+\frac{1\times 37}{3\times 42}\sqrt{109}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Do the multiplications in the fraction \frac{1\times 37}{3\times 42}.

$$\frac{14x^{2}+\frac{37}{3}x-\frac{109}{126}+\frac{37}{126}\sqrt{109}+\frac{37}{3}x+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Combine \frac{37}{3}x and \frac{37}{3}x to get \frac{74}{3}x.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{37}{126}\sqrt{109}+\frac{37}{3}\left(-\frac{1}{42}\right)\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply \frac{37}{3} times -\frac{1}{42} by multiplying numerator times numerator and denominator times denominator.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{37}{126}\sqrt{109}+\frac{37\left(-1\right)}{3\times 42}\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Do the multiplications in the fraction \frac{37\left(-1\right)}{3\times 42}.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{37}{126}\sqrt{109}+\frac{-37}{126}\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Fraction \frac{-37}{126} can be rewritten as -\frac{37}{126} by extracting the negative sign.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{37}{126}\sqrt{109}-\frac{37}{126}\sqrt{109}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Combine \frac{37}{126}\sqrt{109} and -\frac{37}{126}\sqrt{109} to get 0.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{37}{3}\times \frac{37}{42}}{x^{3}-x^{2}-6x}$$

Multiply \frac{37}{3} times \frac{37}{42} by multiplying numerator times numerator and denominator times denominator.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{37\times 37}{3\times 42}}{x^{3}-x^{2}-6x}$$

Do the multiplications in the fraction \frac{37\times 37}{3\times 42}.

$$\frac{14x^{2}+\frac{74}{3}x-\frac{109}{126}+\frac{1369}{126}}{x^{3}-x^{2}-6x}$$

Since -\frac{109}{126} and \frac{1369}{126} have the same denominator, add them by adding their numerators.

$$\frac{14x^{2}+\frac{74}{3}x+\frac{-109+1369}{126}}{x^{3}-x^{2}-6x}$$

Add -109 and 1369 to get 1260.

$$\frac{14x^{2}+\frac{74}{3}x+\frac{1260}{126}}{x^{3}-x^{2}-6x}$$

Divide 1260 by 126 to get 10.

$$\frac{14x^{2}+\frac{74}{3}x+10}{x^{3}-x^{2}-6x}$$

Factor

$$\frac{42\left(x-\frac{-\sqrt{109}-37}{42}\right)\left(x-\frac{\sqrt{109}-37}{42}\right)}{3x\left(x-3\right)\left(x+2\right)}$$