Let S be the set of all real roots of the equation, 3x(3x -- 1) + 2 = |3x -- 1| + |3x -- 2…

Question

Let S be the set of all real roots of the equation,

3x(3x -- 1) + 2 = |3x -- 1| + |3x -- 2|. Then S :

Accepted Answer

Let 3x = t ; t $>$ 0 t(t -- 1) + 2 = |t -- 1| + |t -- 2| t2 -- t + 2 = |t -- 1| + |t -- 2| Case-I : t $<$ 1 t2 -- t + 2 = 1 -- t + 2 -- t $$ \Rightarrow $$ t2 + 2 = 3 -- t $$ \Rightarrow $$ t2 + t -- 1 = 0 $$ \Rightarrow $$ t = $${{ - 1 \pm \sqrt 5 } \over 2}$$ $$ \Rightarrow $$ t = $${{\sqrt 5 - 1} \over 2}$$ [ As t $>$ 0] Case-II : 1 $$ \le $$ t $<$ 2 $$ \Rightarrow $$ t2 -- t + 2 = t -- 1 + 2 -- t $$ \Rightarrow $$ t2 -- t + 1 = 0 D $<$ 0 so no real solution. Case-III : t $$ \ge $$ 2 $$ \Rightarrow $$ t2 -- t + 2 = t -- 1 + t -- 2 $$ \Rightarrow $$ t2 -- 3t - 5 = 0 $$ \Rightarrow $$ D $<$ 0 so no real solution.

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