Solve y^2+5y-6:(y/6-6/y=

$$y ^ { 2 } + 5 y - 6 : ( \frac { y } { 6 } - \frac { 6 } { y } =$$

Evaluate

$$\frac{y\left(y^{3}+5y^{2}-36y-216\right)}{y^{2}-36}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and y is 6y. Multiply \frac{y}{6} times \frac{y}{y}. Multiply \frac{6}{y} times \frac{6}{6}.

$$y^{2}+5y-\frac{6}{\frac{yy}{6y}-\frac{6\times 6}{6y}}$$

Since \frac{yy}{6y} and \frac{6\times 6}{6y} have the same denominator, subtract them by subtracting their numerators.

$$y^{2}+5y-\frac{6}{\frac{yy-6\times 6}{6y}}$$

Do the multiplications in yy-6\times 6.

$$y^{2}+5y-\frac{6}{\frac{y^{2}-36}{6y}}$$

Divide 6 by \frac{y^{2}-36}{6y} by multiplying 6 by the reciprocal of \frac{y^{2}-36}{6y}.

$$y^{2}+5y-\frac{6\times 6y}{y^{2}-36}$$

Multiply 6 and 6 to get 36.

$$y^{2}+5y-\frac{36y}{y^{2}-36}$$

Factor y^{2}-36.

$$y^{2}+5y-\frac{36y}{\left(y-6\right)\left(y+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{2}+5y times \frac{\left(y-6\right)\left(y+6\right)}{\left(y-6\right)\left(y+6\right)}.

$$\frac{\left(y^{2}+5y\right)\left(y-6\right)\left(y+6\right)}{\left(y-6\right)\left(y+6\right)}-\frac{36y}{\left(y-6\right)\left(y+6\right)}$$

Since \frac{\left(y^{2}+5y\right)\left(y-6\right)\left(y+6\right)}{\left(y-6\right)\left(y+6\right)} and \frac{36y}{\left(y-6\right)\left(y+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y^{2}+5y\right)\left(y-6\right)\left(y+6\right)-36y}{\left(y-6\right)\left(y+6\right)}$$

Do the multiplications in \left(y^{2}+5y\right)\left(y-6\right)\left(y+6\right)-36y.

$$\frac{y^{4}-36y^{2}+5y^{3}-180y-36y}{\left(y-6\right)\left(y+6\right)}$$

Combine like terms in y^{4}-36y^{2}+5y^{3}-180y-36y.

$$\frac{y^{4}-36y^{2}+5y^{3}-216y}{\left(y-6\right)\left(y+6\right)}$$

Expand \left(y-6\right)\left(y+6\right).

$$\frac{y^{4}-36y^{2}+5y^{3}-216y}{y^{2}-36}$$

Differentiate w.r.t. y

$$\frac{2y^{5}+5y^{4}-144y^{3}-324y^{2}+2592y+7776}{\left(y^{2}-36\right)^{2}}$$