Solve y-1/2y^2-7y-15+3/y^2-10y+25)divy^2+14/2y-10

$$\frac { y - 1 } { 2 y ^ { 2 } - 7 y - 15 } + \frac { 3 } { y ^ { 2 } - 10 y + 25 } ) \div \frac { y ^ { 2 } + 14 } { 2 y - 10 }$$

Evaluate

$$\frac{y^{2}+14}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Short Solution Steps

Factor 2y^{2}-7y-15. Factor y^{2}-10y+25.

$$\frac{y-1}{\left(y-5\right)\left(2y+3\right)}+\frac{3}{\left(y-5\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-5\right)\left(2y+3\right) and \left(y-5\right)^{2} is \left(2y+3\right)\left(y-5\right)^{2}. Multiply \frac{y-1}{\left(y-5\right)\left(2y+3\right)} times \frac{y-5}{y-5}. Multiply \frac{3}{\left(y-5\right)^{2}} times \frac{2y+3}{2y+3}.

$$\frac{\left(y-1\right)\left(y-5\right)}{\left(2y+3\right)\left(y-5\right)^{2}}+\frac{3\left(2y+3\right)}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Since \frac{\left(y-1\right)\left(y-5\right)}{\left(2y+3\right)\left(y-5\right)^{2}} and \frac{3\left(2y+3\right)}{\left(2y+3\right)\left(y-5\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(y-1\right)\left(y-5\right)+3\left(2y+3\right)}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Do the multiplications in \left(y-1\right)\left(y-5\right)+3\left(2y+3\right).

$$\frac{y^{2}-5y-y+5+6y+9}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Combine like terms in y^{2}-5y-y+5+6y+9.

$$\frac{y^{2}+14}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Expand \left(2y+3\right)\left(y-5\right)^{2}.

$$\frac{y^{2}+14}{2y^{3}-17y^{2}+20y+75}$$

Expand

$$\frac{y^{2}+14}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Short Solution Steps

Factor 2y^{2}-7y-15. Factor y^{2}-10y+25.

$$\frac{y-1}{\left(y-5\right)\left(2y+3\right)}+\frac{3}{\left(y-5\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-5\right)\left(2y+3\right) and \left(y-5\right)^{2} is \left(2y+3\right)\left(y-5\right)^{2}. Multiply \frac{y-1}{\left(y-5\right)\left(2y+3\right)} times \frac{y-5}{y-5}. Multiply \frac{3}{\left(y-5\right)^{2}} times \frac{2y+3}{2y+3}.

$$\frac{\left(y-1\right)\left(y-5\right)}{\left(2y+3\right)\left(y-5\right)^{2}}+\frac{3\left(2y+3\right)}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Since \frac{\left(y-1\right)\left(y-5\right)}{\left(2y+3\right)\left(y-5\right)^{2}} and \frac{3\left(2y+3\right)}{\left(2y+3\right)\left(y-5\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(y-1\right)\left(y-5\right)+3\left(2y+3\right)}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Do the multiplications in \left(y-1\right)\left(y-5\right)+3\left(2y+3\right).

$$\frac{y^{2}-5y-y+5+6y+9}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Combine like terms in y^{2}-5y-y+5+6y+9.

$$\frac{y^{2}+14}{\left(2y+3\right)\left(y-5\right)^{2}}$$

Expand \left(2y+3\right)\left(y-5\right)^{2}.

$$\frac{y^{2}+14}{2y^{3}-17y^{2}+20y+75}$$