Solve 5/6x+18+x+9/2x

$$\frac { 5 } { 6 x + 18 } + \frac { x + 9 } { 2 x }$$

Evaluate

$$\frac{3x^{2}+41x+81}{6x\left(x+3\right)}$$

Short Solution Steps

Factor 6x+18.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6\left(x+3\right) and 2x is 6x\left(x+3\right). Multiply \frac{5}{6\left(x+3\right)} times \frac{x}{x}. Multiply \frac{x+9}{2x} times \frac{3\left(x+3\right)}{3\left(x+3\right)}.

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

Since \frac{5x}{6x\left(x+3\right)} and \frac{\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 5x+\left(x+9\right)\times 3\left(x+3\right).

$$\frac{5x+3x^{2}+9x+27x+81}{6x\left(x+3\right)}$$

Combine like terms in 5x+3x^{2}+9x+27x+81.

$$\frac{41x+3x^{2}+81}{6x\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{41x+3x^{2}+81}{6x\left(x+3\right)}.

$$\frac{3\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{6x\left(x+3\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x\left(x+3\right)}$$

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

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

To find the opposite of -\frac{1}{6}\sqrt{709}-\frac{41}{6}, find the opposite of each term.

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}\right)-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{1}{6}\sqrt{709} is \frac{1}{6}\sqrt{709}.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{41}{6} is \frac{41}{6}.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

To find the opposite of \frac{1}{6}\sqrt{709}-\frac{41}{6}, find the opposite of each term.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{41}{6} is \frac{41}{6}.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)}{2x^{2}+6x}$$

Apply the distributive property by multiplying each term of x+\frac{1}{6}\sqrt{709}+\frac{41}{6} by each term of x-\frac{1}{6}\sqrt{709}+\frac{41}{6}.

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\sqrt{709}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \sqrt{709} and \sqrt{709} to get 709.

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Combine x\left(-\frac{1}{6}\right)\sqrt{709} and \frac{1}{6}\sqrt{709}x to get 0.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{1}{6} and 709 to get \frac{709}{6}.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709}{6}\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{709}{6} times -\frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709\left(-1\right)}{6\times 6}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{709\left(-1\right)}{6\times 6}.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{-709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Fraction \frac{-709}{36} can be rewritten as -\frac{709}{36} by extracting the negative sign.

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{1}{6} times \frac{41}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1\times 41}{6\times 6}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{1\times 41}{6\times 6}.

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Combine x\times \frac{41}{6} and \frac{41}{6}x to get \frac{41}{3}x.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{41}{6} times -\frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41\left(-1\right)}{6\times 6}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{41\left(-1\right)}{6\times 6}.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{-41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Fraction \frac{-41}{36} can be rewritten as -\frac{41}{36} by extracting the negative sign.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}-\frac{41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Combine \frac{41}{36}\sqrt{709} and -\frac{41}{36}\sqrt{709} to get 0.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{41}{6} times \frac{41}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41\times 41}{6\times 6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{41\times 41}{6\times 6}.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{1681}{36}}{2x^{2}+6x}$$

Since -\frac{709}{36} and \frac{1681}{36} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+\frac{41}{3}x+\frac{-709+1681}{36}}{2x^{2}+6x}$$

Add -709 and 1681 to get 972.

$$\frac{x^{2}+\frac{41}{3}x+\frac{972}{36}}{2x^{2}+6x}$$

Divide 972 by 36 to get 27.

$$\frac{x^{2}+\frac{41}{3}x+27}{2x^{2}+6x}$$

Expand

$$\frac{3x^{2}+41x+81}{6x\left(x+3\right)}$$

Short Solution Steps

Factor 6x+18.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6\left(x+3\right) and 2x is 6x\left(x+3\right). Multiply \frac{5}{6\left(x+3\right)} times \frac{x}{x}. Multiply \frac{x+9}{2x} times \frac{3\left(x+3\right)}{3\left(x+3\right)}.

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

Since \frac{5x}{6x\left(x+3\right)} and \frac{\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 5x+\left(x+9\right)\times 3\left(x+3\right).

$$\frac{5x+3x^{2}+9x+27x+81}{6x\left(x+3\right)}$$

Combine like terms in 5x+3x^{2}+9x+27x+81.

$$\frac{41x+3x^{2}+81}{6x\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{41x+3x^{2}+81}{6x\left(x+3\right)}.

$$\frac{3\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{6x\left(x+3\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x\left(x+3\right)}$$

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

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

To find the opposite of -\frac{1}{6}\sqrt{709}-\frac{41}{6}, find the opposite of each term.

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}\right)-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{1}{6}\sqrt{709} is \frac{1}{6}\sqrt{709}.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{41}{6} is \frac{41}{6}.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

To find the opposite of \frac{1}{6}\sqrt{709}-\frac{41}{6}, find the opposite of each term.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{41}{6} is \frac{41}{6}.

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)}{2x^{2}+6x}$$

Apply the distributive property by multiplying each term of x+\frac{1}{6}\sqrt{709}+\frac{41}{6} by each term of x-\frac{1}{6}\sqrt{709}+\frac{41}{6}.

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\sqrt{709}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \sqrt{709} and \sqrt{709} to get 709.

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Combine x\left(-\frac{1}{6}\right)\sqrt{709} and \frac{1}{6}\sqrt{709}x to get 0.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{1}{6} and 709 to get \frac{709}{6}.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709}{6}\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{709}{6} times -\frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709\left(-1\right)}{6\times 6}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{709\left(-1\right)}{6\times 6}.

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{-709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Fraction \frac{-709}{36} can be rewritten as -\frac{709}{36} by extracting the negative sign.

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{1}{6} times \frac{41}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1\times 41}{6\times 6}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{1\times 41}{6\times 6}.

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Combine x\times \frac{41}{6} and \frac{41}{6}x to get \frac{41}{3}x.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{41}{6} times -\frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41\left(-1\right)}{6\times 6}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{41\left(-1\right)}{6\times 6}.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{-41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Fraction \frac{-41}{36} can be rewritten as -\frac{41}{36} by extracting the negative sign.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}-\frac{41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Combine \frac{41}{36}\sqrt{709} and -\frac{41}{36}\sqrt{709} to get 0.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

Multiply \frac{41}{6} times \frac{41}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41\times 41}{6\times 6}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{41\times 41}{6\times 6}.

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{1681}{36}}{2x^{2}+6x}$$

Since -\frac{709}{36} and \frac{1681}{36} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+\frac{41}{3}x+\frac{-709+1681}{36}}{2x^{2}+6x}$$

Add -709 and 1681 to get 972.

$$\frac{x^{2}+\frac{41}{3}x+\frac{972}{36}}{2x^{2}+6x}$$

Divide 972 by 36 to get 27.

$$\frac{x^{2}+\frac{41}{3}x+27}{2x^{2}+6x}$$