Solve 4v/v^2-10v+21-3v/v^2-11v+28

$$\frac { 4 v } { v ^ { 2 } - 10 v + 21 } - \frac { 3 v } { v ^ { 2 } - 11 v + 28 }$$

Evaluate

$$\frac{v}{\left(v-4\right)\left(v-3\right)}$$

Short Solution Steps

Factor v^{2}-10v+21. Factor v^{2}-11v+28.

$$\frac{4v}{\left(v-7\right)\left(v-3\right)}-\frac{3v}{\left(v-7\right)\left(v-4\right)}$$

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

$$\frac{4v\left(v-4\right)}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}-\frac{3v\left(v-3\right)}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}$$

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

$$\frac{4v\left(v-4\right)-3v\left(v-3\right)}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}$$

Do the multiplications in 4v\left(v-4\right)-3v\left(v-3\right).

$$\frac{4v^{2}-16v-3v^{2}+9v}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}$$

Combine like terms in 4v^{2}-16v-3v^{2}+9v.

$$\frac{v^{2}-7v}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}$$

Factor the expressions that are not already factored in \frac{v^{2}-7v}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}.

$$\frac{v\left(v-7\right)}{\left(v-7\right)\left(v-4\right)\left(v-3\right)}$$

Cancel out v-7 in both numerator and denominator.

$$\frac{v}{\left(v-4\right)\left(v-3\right)}$$

Expand \left(v-4\right)\left(v-3\right).

$$\frac{v}{v^{2}-7v+12}$$

Factor

$$\frac{v}{\left(v-4\right)\left(v-3\right)}$$