Solve 3a(x-9a)/x^2-3ax-3a^2-x^2/ax-3a^2

$$\frac { 3 a ( x - 9 a ) } { x ^ { 2 } - 3 a x } - \frac { 3 a ^ { 2 } - x ^ { 2 } } { a x - 3 a ^ { 2 } }$$

Evaluate

$$\frac{x^{2}+3ax+9a^{2}}{ax}$$

Short Solution Steps

Factor x^{2}-3ax. Factor ax-3a^{2}.

$$\frac{3a\left(x-9a\right)}{x\left(x-3a\right)}-\frac{3a^{2}-x^{2}}{a\left(x-3a\right)}$$

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

$$\frac{3a\left(x-9a\right)a}{ax\left(x-3a\right)}-\frac{\left(3a^{2}-x^{2}\right)x}{ax\left(x-3a\right)}$$

Since \frac{3a\left(x-9a\right)a}{ax\left(x-3a\right)} and \frac{\left(3a^{2}-x^{2}\right)x}{ax\left(x-3a\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3a\left(x-9a\right)a-\left(3a^{2}-x^{2}\right)x}{ax\left(x-3a\right)}$$

Do the multiplications in 3a\left(x-9a\right)a-\left(3a^{2}-x^{2}\right)x.

$$\frac{3a^{2}x-27a^{3}-3a^{2}x+x^{3}}{ax\left(x-3a\right)}$$

Combine like terms in 3a^{2}x-27a^{3}-3a^{2}x+x^{3}.

$$\frac{-27a^{3}+x^{3}}{ax\left(x-3a\right)}$$

Factor the expressions that are not already factored in \frac{-27a^{3}+x^{3}}{ax\left(x-3a\right)}.

$$\frac{\left(-x+3a\right)\left(-x^{2}-3ax-9a^{2}\right)}{ax\left(x-3a\right)}$$

Extract the negative sign in 3a-x.

$$\frac{-\left(x-3a\right)\left(-x^{2}-3ax-9a^{2}\right)}{ax\left(x-3a\right)}$$

Cancel out x-3a in both numerator and denominator.

$$\frac{-\left(-x^{2}-3ax-9a^{2}\right)}{ax}$$

To find the opposite of -x^{2}-3ax-9a^{2}, find the opposite of each term.

$$\frac{x^{2}+3ax+9a^{2}}{ax}$$

Expand

$$\frac{x^{2}+3ax+9a^{2}}{ax}$$

Short Solution Steps

Factor x^{2}-3ax. Factor ax-3a^{2}.

$$\frac{3a\left(x-9a\right)}{x\left(x-3a\right)}-\frac{3a^{2}-x^{2}}{a\left(x-3a\right)}$$

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

$$\frac{3a\left(x-9a\right)a}{ax\left(x-3a\right)}-\frac{\left(3a^{2}-x^{2}\right)x}{ax\left(x-3a\right)}$$

Since \frac{3a\left(x-9a\right)a}{ax\left(x-3a\right)} and \frac{\left(3a^{2}-x^{2}\right)x}{ax\left(x-3a\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3a\left(x-9a\right)a-\left(3a^{2}-x^{2}\right)x}{ax\left(x-3a\right)}$$

Do the multiplications in 3a\left(x-9a\right)a-\left(3a^{2}-x^{2}\right)x.

$$\frac{3a^{2}x-27a^{3}-3a^{2}x+x^{3}}{ax\left(x-3a\right)}$$

Combine like terms in 3a^{2}x-27a^{3}-3a^{2}x+x^{3}.

$$\frac{-27a^{3}+x^{3}}{ax\left(x-3a\right)}$$

Factor the expressions that are not already factored in \frac{-27a^{3}+x^{3}}{ax\left(x-3a\right)}.

$$\frac{\left(-x+3a\right)\left(-x^{2}-3ax-9a^{2}\right)}{ax\left(x-3a\right)}$$

Extract the negative sign in 3a-x.

$$\frac{-\left(x-3a\right)\left(-x^{2}-3ax-9a^{2}\right)}{ax\left(x-3a\right)}$$

Cancel out x-3a in both numerator and denominator.

$$\frac{-\left(-x^{2}-3ax-9a^{2}\right)}{ax}$$

To find the opposite of -x^{2}-3ax-9a^{2}, find the opposite of each term.

$$\frac{x^{2}+3ax+9a^{2}}{ax}$$