Solve 7/2a+6-a/10a-8

$$\frac { 7 } { 2 a + 6 } - \frac { a } { 10 a - 8 }$$

Evaluate

$$\frac{-a^{2}+32a-28}{2\left(5a-4\right)\left(a+3\right)}$$

Short Solution Steps

Factor 2a+6. Factor 10a-8.

$$\frac{7}{2\left(a+3\right)}-\frac{a}{2\left(5a-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(a+3\right) and 2\left(5a-4\right) is 2\left(5a-4\right)\left(a+3\right). Multiply \frac{7}{2\left(a+3\right)} times \frac{5a-4}{5a-4}. Multiply \frac{a}{2\left(5a-4\right)} times \frac{a+3}{a+3}.

$$\frac{7\left(5a-4\right)}{2\left(5a-4\right)\left(a+3\right)}-\frac{a\left(a+3\right)}{2\left(5a-4\right)\left(a+3\right)}$$

Since \frac{7\left(5a-4\right)}{2\left(5a-4\right)\left(a+3\right)} and \frac{a\left(a+3\right)}{2\left(5a-4\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(5a-4\right)-a\left(a+3\right)}{2\left(5a-4\right)\left(a+3\right)}$$

Do the multiplications in 7\left(5a-4\right)-a\left(a+3\right).

$$\frac{35a-28-a^{2}-3a}{2\left(5a-4\right)\left(a+3\right)}$$

Combine like terms in 35a-28-a^{2}-3a.

$$\frac{32a-28-a^{2}}{2\left(5a-4\right)\left(a+3\right)}$$

Expand 2\left(5a-4\right)\left(a+3\right).

$$\frac{32a-28-a^{2}}{10a^{2}+22a-24}$$

Factor

$$-\frac{\frac{1}{2}\left(a-\left(16-2\sqrt{57}\right)\right)\left(a-\left(2\sqrt{57}+16\right)\right)}{\left(5a-4\right)\left(a+3\right)}$$