Solve 2p/6p+3/5p^2-23p+12

$$\frac { 2 p } { 6 p } + \frac { 3 } { 5 p ^ { 2 } - 23 p + 12 }$$

Evaluate

$$\frac{5p^{2}-23p+21}{3\left(p-4\right)\left(5p-3\right)}$$

Short Solution Steps

Cancel out 2p in both numerator and denominator.

$$\frac{1}{3}+\frac{3}{5p^{2}-23p+12}$$

Factor 5p^{2}-23p+12.

$$\frac{1}{3}+\frac{3}{\left(p-4\right)\left(5p-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and \left(p-4\right)\left(5p-3\right) is 3\left(p-4\right)\left(5p-3\right). Multiply \frac{1}{3} times \frac{\left(p-4\right)\left(5p-3\right)}{\left(p-4\right)\left(5p-3\right)}. Multiply \frac{3}{\left(p-4\right)\left(5p-3\right)} times \frac{3}{3}.

$$\frac{\left(p-4\right)\left(5p-3\right)}{3\left(p-4\right)\left(5p-3\right)}+\frac{3\times 3}{3\left(p-4\right)\left(5p-3\right)}$$

Since \frac{\left(p-4\right)\left(5p-3\right)}{3\left(p-4\right)\left(5p-3\right)} and \frac{3\times 3}{3\left(p-4\right)\left(5p-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(p-4\right)\left(5p-3\right)+3\times 3}{3\left(p-4\right)\left(5p-3\right)}$$

Do the multiplications in \left(p-4\right)\left(5p-3\right)+3\times 3.

$$\frac{5p^{2}-3p-20p+12+9}{3\left(p-4\right)\left(5p-3\right)}$$

Combine like terms in 5p^{2}-3p-20p+12+9.

$$\frac{5p^{2}-23p+21}{3\left(p-4\right)\left(5p-3\right)}$$

Expand 3\left(p-4\right)\left(5p-3\right).

$$\frac{5p^{2}-23p+21}{15p^{2}-69p+36}$$

Factor

$$\frac{5\left(p-\frac{23-\sqrt{109}}{10}\right)\left(p-\frac{\sqrt{109}+23}{10}\right)}{3\left(p-4\right)\left(5p-3\right)}$$