Solve 9m^2-1/3m^2-6mn+9m^2-6m+1/6m-12n

$$\frac { 9 m ^ { 2 } - 1 } { 3 m ^ { 2 } - 6 m n } + \frac { 9 m ^ { 2 } - 6 m + 1 } { 6 m - 12 n }$$

Evaluate

$$\frac{\left(3m-1\right)\left(m+1\right)\left(3m+2\right)}{6m\left(m-2n\right)}$$

Short Solution Steps

Factor 3m^{2}-6mn. Factor 6m-12n.

$$\frac{9m^{2}-1}{3m\left(m-2n\right)}+\frac{9m^{2}-6m+1}{6\left(m-2n\right)}$$

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

$$\frac{2\left(9m^{2}-1\right)}{6m\left(m-2n\right)}+\frac{\left(9m^{2}-6m+1\right)m}{6m\left(m-2n\right)}$$

Since \frac{2\left(9m^{2}-1\right)}{6m\left(m-2n\right)} and \frac{\left(9m^{2}-6m+1\right)m}{6m\left(m-2n\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(9m^{2}-1\right)+\left(9m^{2}-6m+1\right)m}{6m\left(m-2n\right)}$$

Do the multiplications in 2\left(9m^{2}-1\right)+\left(9m^{2}-6m+1\right)m.

$$\frac{18m^{2}-2+9m^{3}-6m^{2}+m}{6m\left(m-2n\right)}$$

Combine like terms in 18m^{2}-2+9m^{3}-6m^{2}+m.

$$\frac{12m^{2}-2+9m^{3}+m}{6m\left(m-2n\right)}$$

Expand 6m\left(m-2n\right).

$$\frac{12m^{2}-2+9m^{3}+m}{6m^{2}-12mn}$$

Expand

$$-\frac{9m^{3}+12m^{2}+m-2}{6m\left(2n-m\right)}$$

Short Solution Steps

Factor 3m^{2}-6mn. Factor 6m-12n.

$$\frac{9m^{2}-1}{3m\left(m-2n\right)}+\frac{9m^{2}-6m+1}{6\left(m-2n\right)}$$

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

$$\frac{2\left(9m^{2}-1\right)}{6m\left(m-2n\right)}+\frac{\left(9m^{2}-6m+1\right)m}{6m\left(m-2n\right)}$$

Since \frac{2\left(9m^{2}-1\right)}{6m\left(m-2n\right)} and \frac{\left(9m^{2}-6m+1\right)m}{6m\left(m-2n\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(9m^{2}-1\right)+\left(9m^{2}-6m+1\right)m}{6m\left(m-2n\right)}$$

Do the multiplications in 2\left(9m^{2}-1\right)+\left(9m^{2}-6m+1\right)m.

$$\frac{18m^{2}-2+9m^{3}-6m^{2}+m}{6m\left(m-2n\right)}$$

Combine like terms in 18m^{2}-2+9m^{3}-6m^{2}+m.

$$\frac{12m^{2}-2+9m^{3}+m}{6m\left(m-2n\right)}$$

Expand 6m\left(m-2n\right).

$$\frac{12m^{2}-2+9m^{3}+m}{6m^{2}-12mn}$$