Solve H=(y/1-2/5)(2/5+y/3)

$$H = ( \frac { y } { 1 } - \frac { 2 } { 5 } ) ( \frac { 2 } { 5 } + \frac { y } { 3 } )$$

Solve for H

$$H=\frac{y^{2}}{3}+\frac{4y}{15}-\frac{4}{25}$$

Short Solution Steps

Anything divided by one gives itself.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 3 is 15. Multiply \frac{2}{5} times \frac{3}{3}. Multiply \frac{y}{3} times \frac{5}{5}.

$$H=\left(y-\frac{2}{5}\right)\left(\frac{2\times 3}{15}+\frac{5y}{15}\right)$$

Since \frac{2\times 3}{15} and \frac{5y}{15} have the same denominator, add them by adding their numerators.

$$H=\left(y-\frac{2}{5}\right)\times \frac{2\times 3+5y}{15}$$

Do the multiplications in 2\times 3+5y.

$$H=\left(y-\frac{2}{5}\right)\times \frac{6+5y}{15}$$

Use the distributive property to multiply y-\frac{2}{5} by \frac{6+5y}{15}.

$$H=y\times \frac{6+5y}{15}-\frac{2}{5}\times \frac{6+5y}{15}$$

Express y\times \frac{6+5y}{15} as a single fraction.

$$H=\frac{y\left(6+5y\right)}{15}-\frac{2}{5}\times \frac{6+5y}{15}$$

Multiply -\frac{2}{5} times \frac{6+5y}{15} by multiplying numerator times numerator and denominator times denominator.

$$H=\frac{y\left(6+5y\right)}{15}+\frac{-2\left(6+5y\right)}{5\times 15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15 and 5\times 15 is 5\times 15. Multiply \frac{y\left(6+5y\right)}{15} times \frac{5}{5}.

$$H=\frac{5y\left(6+5y\right)}{5\times 15}+\frac{-2\left(6+5y\right)}{5\times 15}$$

Since \frac{5y\left(6+5y\right)}{5\times 15} and \frac{-2\left(6+5y\right)}{5\times 15} have the same denominator, add them by adding their numerators.

$$H=\frac{5y\left(6+5y\right)-2\left(6+5y\right)}{5\times 15}$$

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

$$H=\frac{30y+25y^{2}-12-10y}{5\times 15}$$

Combine like terms in 30y+25y^{2}-12-10y.

$$H=\frac{20y+25y^{2}-12}{5\times 15}$$

Multiply 5 and 15 to get 75.

$$H=\frac{20y+25y^{2}-12}{75}$$

Divide each term of 20y+25y^{2}-12 by 75 to get \frac{4}{15}y+\frac{1}{3}y^{2}-\frac{4}{25}.

$$H=\frac{4}{15}y+\frac{1}{3}y^{2}-\frac{4}{25}$$

Solve for y (complex solution)

$$y=\frac{-\sqrt{75H+16}-2}{5}$$

$$y=\frac{\sqrt{75H+16}-2}{5}$$

Solve for y

$$y=\frac{-\sqrt{75H+16}-2}{5}$$

$$y=\frac{\sqrt{75H+16}-2}{5}\text{, }H\geq -\frac{16}{75}$$