Solve 3x/5x^2+9x-2-6x/5x^2+19x+18

$$\frac { 3 x } { 5 x ^ { 2 } + 9 x - 2 } - \frac { 6 x } { 5 x ^ { 2 } + 19 x + 18 }$$

Evaluate

$$\frac{3x\left(11-5x\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}$$

Short Solution Steps

Factor 5x^{2}+9x-2. Factor 5x^{2}+19x+18.

$$\frac{3x}{\left(5x-1\right)\left(x+2\right)}-\frac{6x}{\left(x+2\right)\left(5x+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(5x-1\right)\left(x+2\right) and \left(x+2\right)\left(5x+9\right) is \left(5x-1\right)\left(x+2\right)\left(5x+9\right). Multiply \frac{3x}{\left(5x-1\right)\left(x+2\right)} times \frac{5x+9}{5x+9}. Multiply \frac{6x}{\left(x+2\right)\left(5x+9\right)} times \frac{5x-1}{5x-1}.

$$\frac{3x\left(5x+9\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}-\frac{6x\left(5x-1\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}$$

Since \frac{3x\left(5x+9\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)} and \frac{6x\left(5x-1\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3x\left(5x+9\right)-6x\left(5x-1\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}$$

Do the multiplications in 3x\left(5x+9\right)-6x\left(5x-1\right).

$$\frac{15x^{2}+27x-30x^{2}+6x}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}$$

Combine like terms in 15x^{2}+27x-30x^{2}+6x.

$$\frac{-15x^{2}+33x}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}$$

Expand \left(5x-1\right)\left(x+2\right)\left(5x+9\right).

$$\frac{-15x^{2}+33x}{25x^{3}+90x^{2}+71x-18}$$

Factor

$$\frac{3x\left(11-5x\right)}{\left(5x-1\right)\left(x+2\right)\left(5x+9\right)}$$