Tractors

Two tractors plow the field in 4 hours together. If the first tractor plow 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 result:

a =  12 h
b =  6 h

Solution:

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

Our quadratic equation calculator calculates it.

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

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