Solve 5/9/x+13/4/x+3+-23/3/(x+3)^2

$$\frac { 5 / 9 } { x } + \frac { 13 / 4 } { x + 3 } + \frac { - 23 / 3 } { ( x + 3 ) ^ { 2 } }$$

Evaluate

$$\frac{137x^{2}+195x+180}{36x\left(x+3\right)^{2}}$$

Short Solution Steps

Express \frac{\frac{5}{9}}{x} as a single fraction.

$$\frac{5}{9x}+\frac{\frac{13}{4}}{x+3}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Express \frac{\frac{13}{4}}{x+3} as a single fraction.

$$\frac{5}{9x}+\frac{13}{4\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9x and 4\left(x+3\right) is 36x\left(x+3\right). Multiply \frac{5}{9x} times \frac{4\left(x+3\right)}{4\left(x+3\right)}. Multiply \frac{13}{4\left(x+3\right)} times \frac{9x}{9x}.

$$\frac{5\times 4\left(x+3\right)}{36x\left(x+3\right)}+\frac{13\times 9x}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Since \frac{5\times 4\left(x+3\right)}{36x\left(x+3\right)} and \frac{13\times 9x}{36x\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\times 4\left(x+3\right)+13\times 9x}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Do the multiplications in 5\times 4\left(x+3\right)+13\times 9x.

$$\frac{20x+60+117x}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Combine like terms in 20x+60+117x.

$$\frac{137x+60}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Use the distributive property to multiply 36x by x+3.

$$\frac{137x+60}{36x^{2}+108x}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Express \frac{-\frac{23}{3}}{\left(x+3\right)^{2}} as a single fraction.

$$\frac{137x+60}{36x^{2}+108x}+\frac{-23}{3\left(x+3\right)^{2}}$$

Factor 36x^{2}+108x.

$$\frac{137x+60}{36x\left(x+3\right)}+\frac{-23}{3\left(x+3\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 36x\left(x+3\right) and 3\left(x+3\right)^{2} is 36x\left(x+3\right)^{2}. Multiply \frac{137x+60}{36x\left(x+3\right)} times \frac{x+3}{x+3}. Multiply \frac{-23}{3\left(x+3\right)^{2}} times \frac{12x}{12x}.

$$\frac{\left(137x+60\right)\left(x+3\right)}{36x\left(x+3\right)^{2}}+\frac{-23\times 12x}{36x\left(x+3\right)^{2}}$$

Since \frac{\left(137x+60\right)\left(x+3\right)}{36x\left(x+3\right)^{2}} and \frac{-23\times 12x}{36x\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(137x+60\right)\left(x+3\right)-23\times 12x}{36x\left(x+3\right)^{2}}$$

Do the multiplications in \left(137x+60\right)\left(x+3\right)-23\times 12x.

$$\frac{137x^{2}+411x+60x+180-276x}{36x\left(x+3\right)^{2}}$$

Combine like terms in 137x^{2}+411x+60x+180-276x.

$$\frac{137x^{2}+195x+180}{36x\left(x+3\right)^{2}}$$

Expand 36x\left(x+3\right)^{2}.

$$\frac{137x^{2}+195x+180}{36x^{3}+216x^{2}+324x}$$

Expand

$$\frac{137x^{2}+195x+180}{36x\left(x+3\right)^{2}}$$

Short Solution Steps

Express \frac{\frac{5}{9}}{x} as a single fraction.

$$\frac{5}{9x}+\frac{\frac{13}{4}}{x+3}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Express \frac{\frac{13}{4}}{x+3} as a single fraction.

$$\frac{5}{9x}+\frac{13}{4\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9x and 4\left(x+3\right) is 36x\left(x+3\right). Multiply \frac{5}{9x} times \frac{4\left(x+3\right)}{4\left(x+3\right)}. Multiply \frac{13}{4\left(x+3\right)} times \frac{9x}{9x}.

$$\frac{5\times 4\left(x+3\right)}{36x\left(x+3\right)}+\frac{13\times 9x}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Since \frac{5\times 4\left(x+3\right)}{36x\left(x+3\right)} and \frac{13\times 9x}{36x\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\times 4\left(x+3\right)+13\times 9x}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Do the multiplications in 5\times 4\left(x+3\right)+13\times 9x.

$$\frac{20x+60+117x}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Combine like terms in 20x+60+117x.

$$\frac{137x+60}{36x\left(x+3\right)}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Use the distributive property to multiply 36x by x+3.

$$\frac{137x+60}{36x^{2}+108x}+\frac{-\frac{23}{3}}{\left(x+3\right)^{2}}$$

Express \frac{-\frac{23}{3}}{\left(x+3\right)^{2}} as a single fraction.

$$\frac{137x+60}{36x^{2}+108x}+\frac{-23}{3\left(x+3\right)^{2}}$$

Factor 36x^{2}+108x.

$$\frac{137x+60}{36x\left(x+3\right)}+\frac{-23}{3\left(x+3\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 36x\left(x+3\right) and 3\left(x+3\right)^{2} is 36x\left(x+3\right)^{2}. Multiply \frac{137x+60}{36x\left(x+3\right)} times \frac{x+3}{x+3}. Multiply \frac{-23}{3\left(x+3\right)^{2}} times \frac{12x}{12x}.

$$\frac{\left(137x+60\right)\left(x+3\right)}{36x\left(x+3\right)^{2}}+\frac{-23\times 12x}{36x\left(x+3\right)^{2}}$$

Since \frac{\left(137x+60\right)\left(x+3\right)}{36x\left(x+3\right)^{2}} and \frac{-23\times 12x}{36x\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(137x+60\right)\left(x+3\right)-23\times 12x}{36x\left(x+3\right)^{2}}$$

Do the multiplications in \left(137x+60\right)\left(x+3\right)-23\times 12x.

$$\frac{137x^{2}+411x+60x+180-276x}{36x\left(x+3\right)^{2}}$$

Combine like terms in 137x^{2}+411x+60x+180-276x.

$$\frac{137x^{2}+195x+180}{36x\left(x+3\right)^{2}}$$

Expand 36x\left(x+3\right)^{2}.

$$\frac{137x^{2}+195x+180}{36x^{3}+216x^{2}+324x}$$