Solve 22x+15/12x^2+52x-9-20-30x/12x^2+52x-9-4-2x/12x^2-52x-9

$$\frac{ 22x+15 }{ 12 { x }^{ 2 } +52x-9 } - \frac{ 20-30x }{ 12 { x }^{ 2 } +52x-9 } - \frac{ 4-2x }{ 12 { x }^{ 2 } -52x-9 }$$

Evaluate

$$\frac{648x^{3}-2708x^{2}-434x+81}{144x^{4}-2920x^{2}+81}$$

Short Solution Steps

Since \frac{22x+15}{12x^{2}+52x-9} and \frac{20-30x}{12x^{2}+52x-9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{22x+15-\left(20-30x\right)}{12x^{2}+52x-9}-\frac{4-2x}{12x^{2}-52x-9}$$

Do the multiplications in 22x+15-\left(20-30x\right).

$$\frac{22x+15-20+30x}{12x^{2}+52x-9}-\frac{4-2x}{12x^{2}-52x-9}$$

Combine like terms in 22x+15-20+30x.

$$\frac{52x-5}{12x^{2}+52x-9}-\frac{4-2x}{12x^{2}-52x-9}$$

Factor 12x^{2}+52x-9. Factor 12x^{2}-52x-9.

$$\frac{52x-5}{\left(6x-1\right)\left(2x+9\right)}-\frac{4-2x}{\left(2x-9\right)\left(6x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(6x-1\right)\left(2x+9\right) and \left(2x-9\right)\left(6x+1\right) is \left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right). Multiply \frac{52x-5}{\left(6x-1\right)\left(2x+9\right)} times \frac{\left(2x-9\right)\left(6x+1\right)}{\left(2x-9\right)\left(6x+1\right)}. Multiply \frac{4-2x}{\left(2x-9\right)\left(6x+1\right)} times \frac{\left(6x-1\right)\left(2x+9\right)}{\left(6x-1\right)\left(2x+9\right)}.

$$\frac{\left(52x-5\right)\left(2x-9\right)\left(6x+1\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}-\frac{\left(4-2x\right)\left(6x-1\right)\left(2x+9\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Since \frac{\left(52x-5\right)\left(2x-9\right)\left(6x+1\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)} and \frac{\left(4-2x\right)\left(6x-1\right)\left(2x+9\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(52x-5\right)\left(2x-9\right)\left(6x+1\right)-\left(4-2x\right)\left(6x-1\right)\left(2x+9\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Do the multiplications in \left(52x-5\right)\left(2x-9\right)\left(6x+1\right)-\left(4-2x\right)\left(6x-1\right)\left(2x+9\right).

$$\frac{624x^{3}-2704x^{2}-468x-60x^{2}+260x+45-48x^{2}-208x+36+24x^{3}+104x^{2}-18x}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Combine like terms in 624x^{3}-2704x^{2}-468x-60x^{2}+260x+45-48x^{2}-208x+36+24x^{3}+104x^{2}-18x.

$$\frac{648x^{3}-2708x^{2}-434x+81}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Expand \left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right).

$$\frac{648x^{3}-2708x^{2}-434x+81}{144x^{4}-2920x^{2}+81}$$

Expand

$$\frac{648x^{3}-2708x^{2}-434x+81}{144x^{4}-2920x^{2}+81}$$

Short Solution Steps

Since \frac{22x+15}{12x^{2}+52x-9} and \frac{20-30x}{12x^{2}+52x-9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{22x+15-\left(20-30x\right)}{12x^{2}+52x-9}-\frac{4-2x}{12x^{2}-52x-9}$$

Do the multiplications in 22x+15-\left(20-30x\right).

$$\frac{22x+15-20+30x}{12x^{2}+52x-9}-\frac{4-2x}{12x^{2}-52x-9}$$

Combine like terms in 22x+15-20+30x.

$$\frac{52x-5}{12x^{2}+52x-9}-\frac{4-2x}{12x^{2}-52x-9}$$

Factor 12x^{2}+52x-9. Factor 12x^{2}-52x-9.

$$\frac{52x-5}{\left(6x-1\right)\left(2x+9\right)}-\frac{4-2x}{\left(2x-9\right)\left(6x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(6x-1\right)\left(2x+9\right) and \left(2x-9\right)\left(6x+1\right) is \left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right). Multiply \frac{52x-5}{\left(6x-1\right)\left(2x+9\right)} times \frac{\left(2x-9\right)\left(6x+1\right)}{\left(2x-9\right)\left(6x+1\right)}. Multiply \frac{4-2x}{\left(2x-9\right)\left(6x+1\right)} times \frac{\left(6x-1\right)\left(2x+9\right)}{\left(6x-1\right)\left(2x+9\right)}.

$$\frac{\left(52x-5\right)\left(2x-9\right)\left(6x+1\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}-\frac{\left(4-2x\right)\left(6x-1\right)\left(2x+9\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Since \frac{\left(52x-5\right)\left(2x-9\right)\left(6x+1\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)} and \frac{\left(4-2x\right)\left(6x-1\right)\left(2x+9\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(52x-5\right)\left(2x-9\right)\left(6x+1\right)-\left(4-2x\right)\left(6x-1\right)\left(2x+9\right)}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Do the multiplications in \left(52x-5\right)\left(2x-9\right)\left(6x+1\right)-\left(4-2x\right)\left(6x-1\right)\left(2x+9\right).

$$\frac{624x^{3}-2704x^{2}-468x-60x^{2}+260x+45-48x^{2}-208x+36+24x^{3}+104x^{2}-18x}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Combine like terms in 624x^{3}-2704x^{2}-468x-60x^{2}+260x+45-48x^{2}-208x+36+24x^{3}+104x^{2}-18x.

$$\frac{648x^{3}-2708x^{2}-434x+81}{\left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right)}$$

Expand \left(2x-9\right)\left(6x-1\right)\left(2x+9\right)\left(6x+1\right).

$$\frac{648x^{3}-2708x^{2}-434x+81}{144x^{4}-2920x^{2}+81}$$