Solve 4(1/x+9-1/x)+4(1/x+9-1/x)+4x(frac{-1}{(x+9)^2}+frac{1}{x^2})

$$4 ( \frac { 1 } { x + 9 } - \frac { 1 } { x } ) + 4 ( \frac { 1 } { x + 9 } - \frac { 1 } { x } ) + 4 x ( \frac { - 1 } { ( x + 9 ) ^ { 2 } } + \frac { 1 } { x ^ { 2 } } )$$

Evaluate

$$-\frac{324}{x\left(x+9\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+9 and x is x\left(x+9\right). Multiply \frac{1}{x+9} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+9}{x+9}.

$$4\left(\frac{x}{x\left(x+9\right)}-\frac{x+9}{x\left(x+9\right)}\right)+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Since \frac{x}{x\left(x+9\right)} and \frac{x+9}{x\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$4\times \frac{x-\left(x+9\right)}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Do the multiplications in x-\left(x+9\right).

$$4\times \frac{x-x-9}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Combine like terms in x-x-9.

$$4\times \frac{-9}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Express 4\times \frac{-9}{x\left(x+9\right)} as a single fraction.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+9 and x is x\left(x+9\right). Multiply \frac{1}{x+9} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+9}{x+9}.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\left(\frac{x}{x\left(x+9\right)}-\frac{x+9}{x\left(x+9\right)}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Since \frac{x}{x\left(x+9\right)} and \frac{x+9}{x\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\times \frac{x-\left(x+9\right)}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Do the multiplications in x-\left(x+9\right).

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\times \frac{x-x-9}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Combine like terms in x-x-9.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\times \frac{-9}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Express 4\times \frac{-9}{x\left(x+9\right)} as a single fraction.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+\frac{4\left(-9\right)}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Combine \frac{4\left(-9\right)}{x\left(x+9\right)} and \frac{4\left(-9\right)}{x\left(x+9\right)} to get 2\times \frac{4\left(-9\right)}{x\left(x+9\right)}.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+9\right)^{2} and x^{2} is x^{2}\left(x+9\right)^{2}. Multiply \frac{-1}{\left(x+9\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{1}{x^{2}} times \frac{\left(x+9\right)^{2}}{\left(x+9\right)^{2}}.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\left(\frac{-x^{2}}{x^{2}\left(x+9\right)^{2}}+\frac{\left(x+9\right)^{2}}{x^{2}\left(x+9\right)^{2}}\right)$$

Since \frac{-x^{2}}{x^{2}\left(x+9\right)^{2}} and \frac{\left(x+9\right)^{2}}{x^{2}\left(x+9\right)^{2}} have the same denominator, add them by adding their numerators.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\times \frac{-x^{2}+\left(x+9\right)^{2}}{x^{2}\left(x+9\right)^{2}}$$

Do the multiplications in -x^{2}+\left(x+9\right)^{2}.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\times \frac{-x^{2}+x^{2}+18x+81}{x^{2}\left(x+9\right)^{2}}$$

Combine like terms in -x^{2}+x^{2}+18x+81.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\times \frac{18x+81}{x^{2}\left(x+9\right)^{2}}$$

Express 4\times \frac{18x+81}{x^{2}\left(x+9\right)^{2}} as a single fraction.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+\frac{4\left(18x+81\right)}{x^{2}\left(x+9\right)^{2}}x$$

Multiply 4 and -9 to get -36.

$$2\times \frac{-36}{x\left(x+9\right)}+\frac{4\left(18x+81\right)}{x^{2}\left(x+9\right)^{2}}x$$

Express 2\times \frac{-36}{x\left(x+9\right)} as a single fraction.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{4\left(18x+81\right)}{x^{2}\left(x+9\right)^{2}}x$$

Use the distributive property to multiply 4 by 18x+81.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{72x+324}{x^{2}\left(x+9\right)^{2}}x$$

Express \frac{72x+324}{x^{2}\left(x+9\right)^{2}}x as a single fraction.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{\left(72x+324\right)x}{x^{2}\left(x+9\right)^{2}}$$

Cancel out x in both numerator and denominator.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{72x+324}{x\left(x+9\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+9\right) and x\left(x+9\right)^{2} is x\left(x+9\right)^{2}. Multiply \frac{2\left(-36\right)}{x\left(x+9\right)} times \frac{x+9}{x+9}.

$$\frac{2\left(-36\right)\left(x+9\right)}{x\left(x+9\right)^{2}}+\frac{72x+324}{x\left(x+9\right)^{2}}$$

Since \frac{2\left(-36\right)\left(x+9\right)}{x\left(x+9\right)^{2}} and \frac{72x+324}{x\left(x+9\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{2\left(-36\right)\left(x+9\right)+72x+324}{x\left(x+9\right)^{2}}$$

Do the multiplications in 2\left(-36\right)\left(x+9\right)+72x+324.

$$\frac{-72x-648+72x+324}{x\left(x+9\right)^{2}}$$

Combine like terms in -72x-648+72x+324.

$$\frac{-324}{x\left(x+9\right)^{2}}$$

Expand x\left(x+9\right)^{2}.

$$\frac{-324}{x^{3}+18x^{2}+81x}$$

Expand

$$-\frac{324}{x\left(x+9\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+9 and x is x\left(x+9\right). Multiply \frac{1}{x+9} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+9}{x+9}.

$$4\left(\frac{x}{x\left(x+9\right)}-\frac{x+9}{x\left(x+9\right)}\right)+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Since \frac{x}{x\left(x+9\right)} and \frac{x+9}{x\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$4\times \frac{x-\left(x+9\right)}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Do the multiplications in x-\left(x+9\right).

$$4\times \frac{x-x-9}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Combine like terms in x-x-9.

$$4\times \frac{-9}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Express 4\times \frac{-9}{x\left(x+9\right)} as a single fraction.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\left(\frac{1}{x+9}-\frac{1}{x}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+9 and x is x\left(x+9\right). Multiply \frac{1}{x+9} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+9}{x+9}.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\left(\frac{x}{x\left(x+9\right)}-\frac{x+9}{x\left(x+9\right)}\right)+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Since \frac{x}{x\left(x+9\right)} and \frac{x+9}{x\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\times \frac{x-\left(x+9\right)}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Do the multiplications in x-\left(x+9\right).

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\times \frac{x-x-9}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Combine like terms in x-x-9.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+4\times \frac{-9}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Express 4\times \frac{-9}{x\left(x+9\right)} as a single fraction.

$$\frac{4\left(-9\right)}{x\left(x+9\right)}+\frac{4\left(-9\right)}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

Combine \frac{4\left(-9\right)}{x\left(x+9\right)} and \frac{4\left(-9\right)}{x\left(x+9\right)} to get 2\times \frac{4\left(-9\right)}{x\left(x+9\right)}.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\left(\frac{-1}{\left(x+9\right)^{2}}+\frac{1}{x^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+9\right)^{2} and x^{2} is x^{2}\left(x+9\right)^{2}. Multiply \frac{-1}{\left(x+9\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{1}{x^{2}} times \frac{\left(x+9\right)^{2}}{\left(x+9\right)^{2}}.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\left(\frac{-x^{2}}{x^{2}\left(x+9\right)^{2}}+\frac{\left(x+9\right)^{2}}{x^{2}\left(x+9\right)^{2}}\right)$$

Since \frac{-x^{2}}{x^{2}\left(x+9\right)^{2}} and \frac{\left(x+9\right)^{2}}{x^{2}\left(x+9\right)^{2}} have the same denominator, add them by adding their numerators.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\times \frac{-x^{2}+\left(x+9\right)^{2}}{x^{2}\left(x+9\right)^{2}}$$

Do the multiplications in -x^{2}+\left(x+9\right)^{2}.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\times \frac{-x^{2}+x^{2}+18x+81}{x^{2}\left(x+9\right)^{2}}$$

Combine like terms in -x^{2}+x^{2}+18x+81.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+4x\times \frac{18x+81}{x^{2}\left(x+9\right)^{2}}$$

Express 4\times \frac{18x+81}{x^{2}\left(x+9\right)^{2}} as a single fraction.

$$2\times \frac{4\left(-9\right)}{x\left(x+9\right)}+\frac{4\left(18x+81\right)}{x^{2}\left(x+9\right)^{2}}x$$

Multiply 4 and -9 to get -36.

$$2\times \frac{-36}{x\left(x+9\right)}+\frac{4\left(18x+81\right)}{x^{2}\left(x+9\right)^{2}}x$$

Express 2\times \frac{-36}{x\left(x+9\right)} as a single fraction.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{4\left(18x+81\right)}{x^{2}\left(x+9\right)^{2}}x$$

Use the distributive property to multiply 4 by 18x+81.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{72x+324}{x^{2}\left(x+9\right)^{2}}x$$

Express \frac{72x+324}{x^{2}\left(x+9\right)^{2}}x as a single fraction.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{\left(72x+324\right)x}{x^{2}\left(x+9\right)^{2}}$$

Cancel out x in both numerator and denominator.

$$\frac{2\left(-36\right)}{x\left(x+9\right)}+\frac{72x+324}{x\left(x+9\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+9\right) and x\left(x+9\right)^{2} is x\left(x+9\right)^{2}. Multiply \frac{2\left(-36\right)}{x\left(x+9\right)} times \frac{x+9}{x+9}.

$$\frac{2\left(-36\right)\left(x+9\right)}{x\left(x+9\right)^{2}}+\frac{72x+324}{x\left(x+9\right)^{2}}$$

Since \frac{2\left(-36\right)\left(x+9\right)}{x\left(x+9\right)^{2}} and \frac{72x+324}{x\left(x+9\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{2\left(-36\right)\left(x+9\right)+72x+324}{x\left(x+9\right)^{2}}$$

Do the multiplications in 2\left(-36\right)\left(x+9\right)+72x+324.

$$\frac{-72x-648+72x+324}{x\left(x+9\right)^{2}}$$

Combine like terms in -72x-648+72x+324.

$$\frac{-324}{x\left(x+9\right)^{2}}$$

Expand x\left(x+9\right)^{2}.

$$\frac{-324}{x^{3}+18x^{2}+81x}$$