Solve 5/6x+18+x+4/2x

$$\frac{ 5 }{ 6x+18 } + \frac{ x+4 }{ 2x }$$

Evaluate

$$\frac{3x^{2}+26x+36}{6x\left(x+3\right)}$$

Short Solution Steps

Factor 6x+18.

$$\frac{5}{6\left(x+3\right)}+\frac{x+4}{2x}$$

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+4}{2x} times \frac{3\left(x+3\right)}{3\left(x+3\right)}.

$$\frac{5x}{6x\left(x+3\right)}+\frac{\left(x+4\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

Since \frac{5x}{6x\left(x+3\right)} and \frac{\left(x+4\right)\times 3\left(x+3\right)}{6x\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5x+\left(x+4\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

Do the multiplications in 5x+\left(x+4\right)\times 3\left(x+3\right).

$$\frac{5x+3x^{2}+9x+12x+36}{6x\left(x+3\right)}$$

Combine like terms in 5x+3x^{2}+9x+12x+36.

$$\frac{26x+3x^{2}+36}{6x\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{26x+3x^{2}+36}{6x\left(x+3\right)}.

$$\frac{3\left(x-\left(-\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{6x\left(x+3\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x\left(x+3\right)}$$

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

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

To find the opposite of -\frac{1}{3}\sqrt{61}-\frac{13}{3}, find the opposite of each term.

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{61}\right)-\left(-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{1}{3}\sqrt{61} is \frac{1}{3}\sqrt{61}.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}-\left(-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{13}{3} is \frac{13}{3}.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

To find the opposite of \frac{1}{3}\sqrt{61}-\frac{13}{3}, find the opposite of each term.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)\left(x-\frac{1}{3}\sqrt{61}-\left(-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{13}{3} is \frac{13}{3}.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)\left(x-\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)}{2x^{2}+6x}$$

Apply the distributive property by multiplying each term of x+\frac{1}{3}\sqrt{61}+\frac{13}{3} by each term of x-\frac{1}{3}\sqrt{61}+\frac{13}{3}.

$$\frac{x^{2}+x\left(-\frac{1}{3}\right)\sqrt{61}+x\times \frac{13}{3}+\frac{1}{3}\sqrt{61}x+\frac{1}{3}\sqrt{61}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \sqrt{61} and \sqrt{61} to get 61.

$$\frac{x^{2}+x\left(-\frac{1}{3}\right)\sqrt{61}+x\times \frac{13}{3}+\frac{1}{3}\sqrt{61}x+\frac{1}{3}\times 61\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Combine x\left(-\frac{1}{3}\right)\sqrt{61} and \frac{1}{3}\sqrt{61}x to get 0.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{1}{3}\times 61\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{1}{3} and 61 to get \frac{61}{3}.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{61}{3}\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{61}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{61\left(-1\right)}{3\times 3}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{61\left(-1\right)}{3\times 3}.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{-61}{9}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Fraction \frac{-61}{9} can be rewritten as -\frac{61}{9} by extracting the negative sign.

$$\frac{x^{2}+x\times \frac{13}{3}-\frac{61}{9}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{1}{3} times \frac{13}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{13}{3}-\frac{61}{9}+\frac{1\times 13}{3\times 3}\sqrt{61}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{1\times 13}{3\times 3}.

$$\frac{x^{2}+x\times \frac{13}{3}-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Combine x\times \frac{13}{3} and \frac{13}{3}x to get \frac{26}{3}x.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{13}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{13\left(-1\right)}{3\times 3}\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{13\left(-1\right)}{3\times 3}.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{-13}{9}\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Fraction \frac{-13}{9} can be rewritten as -\frac{13}{9} by extracting the negative sign.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}-\frac{13}{9}\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Combine \frac{13}{9}\sqrt{61} and -\frac{13}{9}\sqrt{61} to get 0.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{13}{3} times \frac{13}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13\times 13}{3\times 3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{13\times 13}{3\times 3}.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{169}{9}}{2x^{2}+6x}$$

Since -\frac{61}{9} and \frac{169}{9} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+\frac{26}{3}x+\frac{-61+169}{9}}{2x^{2}+6x}$$

Add -61 and 169 to get 108.

$$\frac{x^{2}+\frac{26}{3}x+\frac{108}{9}}{2x^{2}+6x}$$

Divide 108 by 9 to get 12.

$$\frac{x^{2}+\frac{26}{3}x+12}{2x^{2}+6x}$$

Expand

$$\frac{3x^{2}+26x+36}{6x\left(x+3\right)}$$

Short Solution Steps

Factor 6x+18.

$$\frac{5}{6\left(x+3\right)}+\frac{x+4}{2x}$$

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+4}{2x} times \frac{3\left(x+3\right)}{3\left(x+3\right)}.

$$\frac{5x}{6x\left(x+3\right)}+\frac{\left(x+4\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

Since \frac{5x}{6x\left(x+3\right)} and \frac{\left(x+4\right)\times 3\left(x+3\right)}{6x\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5x+\left(x+4\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

Do the multiplications in 5x+\left(x+4\right)\times 3\left(x+3\right).

$$\frac{5x+3x^{2}+9x+12x+36}{6x\left(x+3\right)}$$

Combine like terms in 5x+3x^{2}+9x+12x+36.

$$\frac{26x+3x^{2}+36}{6x\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{26x+3x^{2}+36}{6x\left(x+3\right)}.

$$\frac{3\left(x-\left(-\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{6x\left(x+3\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x\left(x+3\right)}$$

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

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

To find the opposite of -\frac{1}{3}\sqrt{61}-\frac{13}{3}, find the opposite of each term.

$$\frac{\left(x-\left(-\frac{1}{3}\sqrt{61}\right)-\left(-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{1}{3}\sqrt{61} is \frac{1}{3}\sqrt{61}.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}-\left(-\frac{13}{3}\right)\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{13}{3} is \frac{13}{3}.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)\left(x-\left(\frac{1}{3}\sqrt{61}-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

To find the opposite of \frac{1}{3}\sqrt{61}-\frac{13}{3}, find the opposite of each term.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)\left(x-\frac{1}{3}\sqrt{61}-\left(-\frac{13}{3}\right)\right)}{2x^{2}+6x}$$

The opposite of -\frac{13}{3} is \frac{13}{3}.

$$\frac{\left(x+\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)\left(x-\frac{1}{3}\sqrt{61}+\frac{13}{3}\right)}{2x^{2}+6x}$$

Apply the distributive property by multiplying each term of x+\frac{1}{3}\sqrt{61}+\frac{13}{3} by each term of x-\frac{1}{3}\sqrt{61}+\frac{13}{3}.

$$\frac{x^{2}+x\left(-\frac{1}{3}\right)\sqrt{61}+x\times \frac{13}{3}+\frac{1}{3}\sqrt{61}x+\frac{1}{3}\sqrt{61}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \sqrt{61} and \sqrt{61} to get 61.

$$\frac{x^{2}+x\left(-\frac{1}{3}\right)\sqrt{61}+x\times \frac{13}{3}+\frac{1}{3}\sqrt{61}x+\frac{1}{3}\times 61\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Combine x\left(-\frac{1}{3}\right)\sqrt{61} and \frac{1}{3}\sqrt{61}x to get 0.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{1}{3}\times 61\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{1}{3} and 61 to get \frac{61}{3}.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{61}{3}\left(-\frac{1}{3}\right)+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{61}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{61\left(-1\right)}{3\times 3}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{61\left(-1\right)}{3\times 3}.

$$\frac{x^{2}+x\times \frac{13}{3}+\frac{-61}{9}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Fraction \frac{-61}{9} can be rewritten as -\frac{61}{9} by extracting the negative sign.

$$\frac{x^{2}+x\times \frac{13}{3}-\frac{61}{9}+\frac{1}{3}\sqrt{61}\times \frac{13}{3}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{1}{3} times \frac{13}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+x\times \frac{13}{3}-\frac{61}{9}+\frac{1\times 13}{3\times 3}\sqrt{61}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{1\times 13}{3\times 3}.

$$\frac{x^{2}+x\times \frac{13}{3}-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{13}{3}x+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Combine x\times \frac{13}{3} and \frac{13}{3}x to get \frac{26}{3}x.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{13}{3}\left(-\frac{1}{3}\right)\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{13}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{13\left(-1\right)}{3\times 3}\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{13\left(-1\right)}{3\times 3}.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}+\frac{-13}{9}\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Fraction \frac{-13}{9} can be rewritten as -\frac{13}{9} by extracting the negative sign.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{9}\sqrt{61}-\frac{13}{9}\sqrt{61}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Combine \frac{13}{9}\sqrt{61} and -\frac{13}{9}\sqrt{61} to get 0.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13}{3}\times \frac{13}{3}}{2x^{2}+6x}$$

Multiply \frac{13}{3} times \frac{13}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{13\times 13}{3\times 3}}{2x^{2}+6x}$$

Do the multiplications in the fraction \frac{13\times 13}{3\times 3}.

$$\frac{x^{2}+\frac{26}{3}x-\frac{61}{9}+\frac{169}{9}}{2x^{2}+6x}$$

Since -\frac{61}{9} and \frac{169}{9} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+\frac{26}{3}x+\frac{-61+169}{9}}{2x^{2}+6x}$$

Add -61 and 169 to get 108.

$$\frac{x^{2}+\frac{26}{3}x+\frac{108}{9}}{2x^{2}+6x}$$

Divide 108 by 9 to get 12.

$$\frac{x^{2}+\frac{26}{3}x+12}{2x^{2}+6x}$$