Car factory

Carmaker now produce 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 needed to manufacture 70312 cars last year?

Correct result:

w =  188 d

Solution:

$(x+2) \cdot \ (w-1)=70312 \ \\ w x=70312 \ \\ \ \\ (70312/w+2) \cdot \ (w-1)=70312 \ \\ \ \\ (70312+2*w)*(w-1)=70312*w \ \\ \ \\ (70312+2 \cdot \ w) \cdot \ (w-1)=70312 \cdot \ w \ \\ 2w^2 -2w -70312=0 \ \\ \ \\ a=2; b=-2; c=-70312 \ \\ D=b^2 - 4ac=2^2 - 4\cdot 2 \cdot (-70312)=562500 \ \\ D>0 \ \\ \ \\ w_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 2 \pm \sqrt{ 562500 } }{ 4 } \ \\ w_{1,2}=\dfrac{ 2 \pm 750 }{ 4 } \ \\ w_{1,2}=0.5 \pm 187.5 \ \\ w_{1}=188 \ \\ w_{2}=-187 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 2 (w -188) (w +187)=0 \ \\ \ \\ w=w_{1}=188 \ \\ x_{1}=70312/w=70312/188=374 \ \text{a/d} \ \\ w=w_{1}=188 \ \text{d}$

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