Solve y^3-125/y^2-25*y+5/6y=y^2+5y+25/6y

$$\frac { y ^ { 3 } - 125 } { y ^ { 2 } - 25 } \cdot \frac { y + 5 } { 6 y } = \frac { y ^ { 2 } + 5 y + 25 } { 6 y }$$

Solve for y (complex solution)

$$y\in \mathrm{C}\setminus -5,5,0$$

Short Solution Steps

Variable y cannot be equal to any of the values -5,0,5 since division by zero is not defined. Multiply both sides of the equation by 6y\left(y-5\right)\left(y+5\right), the least common multiple of y^{2}-25,6y.

$$6y\left(y^{3}-125\right)\times \frac{y+5}{6y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express 6\times \frac{y+5}{6y} as a single fraction.

$$\frac{6\left(y+5\right)}{6y}y\left(y^{3}-125\right)=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Cancel out 6 in both numerator and denominator.

$$\frac{y+5}{y}y\left(y^{3}-125\right)=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Use the distributive property to multiply \frac{y+5}{y}y by y^{3}-125.

$$\frac{y+5}{y}y^{4}-125\times \frac{y+5}{y}y=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express \frac{y+5}{y}y^{4} as a single fraction.

$$\frac{\left(y+5\right)y^{4}}{y}-125\times \frac{y+5}{y}y=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express -125\times \frac{y+5}{y} as a single fraction.

$$\frac{\left(y+5\right)y^{4}}{y}+\frac{-125\left(y+5\right)}{y}y=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express \frac{-125\left(y+5\right)}{y}y as a single fraction.

$$\frac{\left(y+5\right)y^{4}}{y}+\frac{-125\left(y+5\right)y}{y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Since \frac{\left(y+5\right)y^{4}}{y} and \frac{-125\left(y+5\right)y}{y} have the same denominator, add them by adding their numerators.

$$\frac{\left(y+5\right)y^{4}-125\left(y+5\right)y}{y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Do the multiplications in \left(y+5\right)y^{4}-125\left(y+5\right)y.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Use the distributive property to multiply y^{2}-25 by y^{2}+5y+25 and combine like terms.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}=y^{4}+5y^{3}-125y-625$$

Subtract y^{4} from both sides.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}-y^{4}=5y^{3}-125y-625$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{4} times \frac{y}{y}.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}-\frac{y^{4}y}{y}=5y^{3}-125y-625$$

Since \frac{y^{5}+5y^{4}-125y^{2}-625y}{y} and \frac{y^{4}y}{y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y-y^{4}y}{y}=5y^{3}-125y-625$$

Do the multiplications in y^{5}+5y^{4}-125y^{2}-625y-y^{4}y.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y-y^{5}}{y}=5y^{3}-125y-625$$

Combine like terms in y^{5}+5y^{4}-125y^{2}-625y-y^{5}.

$$\frac{5y^{4}-125y^{2}-625y}{y}=5y^{3}-125y-625$$

Subtract 5y^{3} from both sides.

$$\frac{5y^{4}-125y^{2}-625y}{y}-5y^{3}=-125y-625$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -5y^{3} times \frac{y}{y}.

$$\frac{5y^{4}-125y^{2}-625y}{y}+\frac{-5y^{3}y}{y}=-125y-625$$

Since \frac{5y^{4}-125y^{2}-625y}{y} and \frac{-5y^{3}y}{y} have the same denominator, add them by adding their numerators.

$$\frac{5y^{4}-125y^{2}-625y-5y^{3}y}{y}=-125y-625$$

Do the multiplications in 5y^{4}-125y^{2}-625y-5y^{3}y.

$$\frac{5y^{4}-125y^{2}-625y-5y^{4}}{y}=-125y-625$$

Combine like terms in 5y^{4}-125y^{2}-625y-5y^{4}.

$$\frac{-125y^{2}-625y}{y}=-125y-625$$

Add 125y to both sides.

$$\frac{-125y^{2}-625y}{y}+125y=-625$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 125y times \frac{y}{y}.

$$\frac{-125y^{2}-625y}{y}+\frac{125yy}{y}=-625$$

Since \frac{-125y^{2}-625y}{y} and \frac{125yy}{y} have the same denominator, add them by adding their numerators.

$$\frac{-125y^{2}-625y+125yy}{y}=-625$$

Do the multiplications in -125y^{2}-625y+125yy.

$$\frac{-125y^{2}-625y+125y^{2}}{y}=-625$$

Combine like terms in -125y^{2}-625y+125y^{2}.

$$\frac{-625y}{y}=-625$$

Add 625 to both sides.

$$\frac{-625y}{y}+625=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 625 times \frac{y}{y}.

$$\frac{-625y}{y}+\frac{625y}{y}=0$$

Since \frac{-625y}{y} and \frac{625y}{y} have the same denominator, add them by adding their numerators.

$$\frac{-625y+625y}{y}=0$$

Combine like terms in -625y+625y.

$$\frac{0}{y}=0$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$0=0$$

This is true for any y.

$$y\in \mathrm{C}$$

Variable y cannot be equal to any of the values 0,-5,5.

$$y\in \mathrm{C}\setminus -5,0,5$$

Solve for y

$$y\in \mathrm{R}\setminus 5,-5,0$$

Short Solution Steps

Variable y cannot be equal to any of the values -5,0,5 since division by zero is not defined. Multiply both sides of the equation by 6y\left(y-5\right)\left(y+5\right), the least common multiple of y^{2}-25,6y.

$$6y\left(y^{3}-125\right)\times \frac{y+5}{6y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express 6\times \frac{y+5}{6y} as a single fraction.

$$\frac{6\left(y+5\right)}{6y}y\left(y^{3}-125\right)=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Cancel out 6 in both numerator and denominator.

$$\frac{y+5}{y}y\left(y^{3}-125\right)=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Use the distributive property to multiply \frac{y+5}{y}y by y^{3}-125.

$$\frac{y+5}{y}y^{4}-125\times \frac{y+5}{y}y=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express \frac{y+5}{y}y^{4} as a single fraction.

$$\frac{\left(y+5\right)y^{4}}{y}-125\times \frac{y+5}{y}y=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express -125\times \frac{y+5}{y} as a single fraction.

$$\frac{\left(y+5\right)y^{4}}{y}+\frac{-125\left(y+5\right)}{y}y=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Express \frac{-125\left(y+5\right)}{y}y as a single fraction.

$$\frac{\left(y+5\right)y^{4}}{y}+\frac{-125\left(y+5\right)y}{y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Since \frac{\left(y+5\right)y^{4}}{y} and \frac{-125\left(y+5\right)y}{y} have the same denominator, add them by adding their numerators.

$$\frac{\left(y+5\right)y^{4}-125\left(y+5\right)y}{y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Do the multiplications in \left(y+5\right)y^{4}-125\left(y+5\right)y.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}=\left(y^{2}-25\right)\left(y^{2}+5y+25\right)$$

Use the distributive property to multiply y^{2}-25 by y^{2}+5y+25 and combine like terms.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}=y^{4}+5y^{3}-125y-625$$

Subtract y^{4} from both sides.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}-y^{4}=5y^{3}-125y-625$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{4} times \frac{y}{y}.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y}{y}-\frac{y^{4}y}{y}=5y^{3}-125y-625$$

Since \frac{y^{5}+5y^{4}-125y^{2}-625y}{y} and \frac{y^{4}y}{y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y-y^{4}y}{y}=5y^{3}-125y-625$$

Do the multiplications in y^{5}+5y^{4}-125y^{2}-625y-y^{4}y.

$$\frac{y^{5}+5y^{4}-125y^{2}-625y-y^{5}}{y}=5y^{3}-125y-625$$

Combine like terms in y^{5}+5y^{4}-125y^{2}-625y-y^{5}.

$$\frac{5y^{4}-125y^{2}-625y}{y}=5y^{3}-125y-625$$

Subtract 5y^{3} from both sides.

$$\frac{5y^{4}-125y^{2}-625y}{y}-5y^{3}=-125y-625$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -5y^{3} times \frac{y}{y}.

$$\frac{5y^{4}-125y^{2}-625y}{y}+\frac{-5y^{3}y}{y}=-125y-625$$

Since \frac{5y^{4}-125y^{2}-625y}{y} and \frac{-5y^{3}y}{y} have the same denominator, add them by adding their numerators.

$$\frac{5y^{4}-125y^{2}-625y-5y^{3}y}{y}=-125y-625$$

Do the multiplications in 5y^{4}-125y^{2}-625y-5y^{3}y.

$$\frac{5y^{4}-125y^{2}-625y-5y^{4}}{y}=-125y-625$$

Combine like terms in 5y^{4}-125y^{2}-625y-5y^{4}.

$$\frac{-125y^{2}-625y}{y}=-125y-625$$

Add 125y to both sides.

$$\frac{-125y^{2}-625y}{y}+125y=-625$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 125y times \frac{y}{y}.

$$\frac{-125y^{2}-625y}{y}+\frac{125yy}{y}=-625$$

Since \frac{-125y^{2}-625y}{y} and \frac{125yy}{y} have the same denominator, add them by adding their numerators.

$$\frac{-125y^{2}-625y+125yy}{y}=-625$$

Do the multiplications in -125y^{2}-625y+125yy.

$$\frac{-125y^{2}-625y+125y^{2}}{y}=-625$$

Combine like terms in -125y^{2}-625y+125y^{2}.

$$\frac{-625y}{y}=-625$$

Add 625 to both sides.

$$\frac{-625y}{y}+625=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 625 times \frac{y}{y}.

$$\frac{-625y}{y}+\frac{625y}{y}=0$$

Since \frac{-625y}{y} and \frac{625y}{y} have the same denominator, add them by adding their numerators.

$$\frac{-625y+625y}{y}=0$$

Combine like terms in -625y+625y.

$$\frac{0}{y}=0$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$0=0$$

This is true for any y.

$$y\in \mathrm{R}$$

Variable y cannot be equal to any of the values 0,-5,5.

$$y\in \mathrm{R}\setminus -5,0,5$$