Mountain climbing

Ken and his brother decided to go on mountain climbing 8 miles from their house to Mt. Daraitan at a rate of x mph (miles per hour). For the return trip, it was 2 mph faster. It took them 6 hours for the entire round trip. What is the x?

Correct answer:

x =  2 mph

Step-by-step explanation:

$$\begin{gathered} s=8\text{mi} \\ t=6\text{h} \\ s=x\cdot t_1 \\ s=(x+2)\cdot t_2 \\ t=t_1+t_2 \\ 8=x\cdot t_1 \\ 8=(x+2)\cdot (6-t_1) \\ 8=(x+2)\cdot (6-8/x) \\ 8\cdot x=(x+2)\cdot (6x-8) \\ 8\cdot x=(x+2)\cdot (6x-8) \\ -6x^2+4x+16=0 \\ 6x^2-4x-16=0 \\ 6=2\cdot 3 \\ 4=2^2 \\ 16=2^4 \\ \text{GCD}(6,4,16)=2 \\ 3x^2-2x-8=0 \\ a=3;b=-2;c=-8 \\ D=b^2-4ac=2^2-4\cdot 3\cdot (-8)=100 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{2\pm \sqrt{100}}{6} \\ {x}_{1,2}=\frac{2\pm 10}{6} \\ {x}_{1,2}=0.333333\pm 1.666667 \\ {x}_1=2 \\ {x}_2=-1.333333333 \\ x=x_1=2=2\text{mph} \\ \text{ Verifying Solution: } \\ {t}_1=s/x=8/2=4\text{h} \\ {t}_2=\frac{s}{x+2}=\frac{8}{2+2}=2\text{h} \\ {t}_3=t_1+t_2=4+2=6\text{h} \end{gathered}$$

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