Solve y+7/y^2-7y+10-7/y^2-25

$$\frac { y + 7 } { y ^ { 2 } - 7 y + 10 } - \frac { 7 } { y ^ { 2 } - 25 }$$

Evaluate

$$\frac{y^{2}+5y+49}{\left(y-2\right)\left(y^{2}-25\right)}$$

Short Solution Steps

Factor y^{2}-7y+10. Factor y^{2}-25.

$$\frac{y+7}{\left(y-5\right)\left(y-2\right)}-\frac{7}{\left(y-5\right)\left(y+5\right)}$$

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

$$\frac{\left(y+7\right)\left(y+5\right)}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}-\frac{7\left(y-2\right)}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

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

$$\frac{\left(y+7\right)\left(y+5\right)-7\left(y-2\right)}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

Do the multiplications in \left(y+7\right)\left(y+5\right)-7\left(y-2\right).

$$\frac{y^{2}+5y+7y+35-7y+14}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

Combine like terms in y^{2}+5y+7y+35-7y+14.

$$\frac{y^{2}+5y+49}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

Expand \left(y-5\right)\left(y-2\right)\left(y+5\right).

$$\frac{y^{2}+5y+49}{y^{3}-2y^{2}-25y+50}$$

Expand

$$\frac{y^{2}+5y+49}{\left(y-2\right)\left(y^{2}-25\right)}$$

Short Solution Steps

Factor y^{2}-7y+10. Factor y^{2}-25.

$$\frac{y+7}{\left(y-5\right)\left(y-2\right)}-\frac{7}{\left(y-5\right)\left(y+5\right)}$$

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

$$\frac{\left(y+7\right)\left(y+5\right)}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}-\frac{7\left(y-2\right)}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

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

$$\frac{\left(y+7\right)\left(y+5\right)-7\left(y-2\right)}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

Do the multiplications in \left(y+7\right)\left(y+5\right)-7\left(y-2\right).

$$\frac{y^{2}+5y+7y+35-7y+14}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

Combine like terms in y^{2}+5y+7y+35-7y+14.

$$\frac{y^{2}+5y+49}{\left(y-5\right)\left(y-2\right)\left(y+5\right)}$$

Expand \left(y-5\right)\left(y-2\right)\left(y+5\right).

$$\frac{y^{2}+5y+49}{y^{3}-2y^{2}-25y+50}$$