Car factory

The carmaker now produces 2 cars a day more than last year, so the production of 70312 cars will save just one full working day. How many working days were needed to manufacture 70312 cars last year?

Correct answer:

w =  188 d

Step-by-step explanation:

$$\begin{gathered} (x+2)\cdot (w-1)=70312 \\ wx=70312 \\ (70312/w+2)\cdot (w-1)=70312 \\ (70312+2\cdot w)\cdot (w-1)=70312\cdot w \\ (70312+2\cdot w)\cdot (w-1)=70312\cdot w \\ 2w^2-2w-70312=0 \\ 2...\text{ prime number} \\ 70312=2^3\cdot 11\cdot 17\cdot 47 \\ \text{GCD}(2,2,70312)=2 \\ {w}^2-w-35156=0 \\ a=1;b=-1;c=-35156 \\ D=b^2-4ac=1^2-4\cdot 1\cdot (-35156)=140625 \\ D>0 \\ {w}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{1\pm \sqrt{140625}}{2} \\ {w}_{1,2}=\frac{1\pm 375}{2} \\ {w}_{1,2}=0.5\pm 187.5 \\ {w}_1=188 \\ {w}_2=-187 \\ w=w_1=188 \\ {x}_1=70312/w=70312/188=374\text{a/d} \\ w=w_1=188=188\text{d} \end{gathered}$$

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