Tractors

Two tractors plow the field for 4 hours together. If the first tractor plowed half of the field and then the second tractor completed the job, it would take 9 hours. How many hours does the field plow for each tractor separately?

Correct answer:

a =  12 h
b =  6 h

Step-by-step explanation:

$$\begin{gathered} 4\cdot (\frac{1}{a}+\frac{1}{b})=1 \\ a/2+b/2=9 \\ a+b=18 \\ b=18-a \\ 4\cdot (b+a)=ab \\ 4\cdot ((18-a)+a)=a(18-a) \\ 4\cdot ((18-a)+a)=a(18-a) \\ {a}^2-18a+72=0 \\ p=1;q=-18;r=72 \\ D=q^2-4pr=18^2-4\cdot 1\cdot 72=36 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{18\pm \sqrt{36}}{2} \\ {a}_{1,2}=\frac{18\pm 6}{2} \\ {a}_{1,2}=9\pm 3 \\ {a}_1=12 \\ {a}_2=6 \\ a=a_1=12=12\text{h} \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=18-a=18-12=6\text{h}$

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