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The red hose takes 3 hours more than the green hose to fill the hot tub . Together, both hoses fills the hot tub in 2 hours. Find how long does it take for each hose to fill the hot tub.

Consider x be the time needed to fill the hot tub for green hose.

The time needed to fill the hot tub for red hose is $x+3$.

Form a table using given the information-

Hot tub filled can be express in equation below

$\frac{1}{2}=\frac{1}{x}+\frac{1}{x+3}$

LCD of denominator is $2x\left(x+3\right)$

Multiply both sides by the LCD.

$2x\left(x+3\right)\frac{1}{2}=2x\left(x+3\right)\left(\frac{1}{x}+\frac{1}{x+3}\right)$

Simplify the equation

$$\begin{gathered} x\left(x+3\right)=2\left(x+3\right)+2x \\ {x}^2+3x=2x+6+2x \end{gathered}$$

Combine like terms

$x^2-x-6=0$

Solve the quadratic equation

$\left(x-3\right)\left(x+2\right)=0$

So,

$x-3=0⇒x=3$

$x+2=0⇒x=-2$

Number of hours can not be negative.

The time needed to fill the hot tub by green hose is 3 hours.

The time needed to fill the hot tub by red hose is 6 hours.