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Question

For xR, the number of real roots of the equation 3x24|x21|+x1=0 is

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Solution

Given:
3x24|x21|+x1=0
Case 1: x21<0x(1,1)
3x2+4(x21)+x1=0
7x2+x5=0
x=1±14114
x=114114,1+14114
Both roots lie in (1,1)

Case 2: x210x(,1][1,)
3x24(x21)+x1=0
x2x3=0
x=1±132
x=1132,1+132
Both roots lie in (,1][1,)
Hence, total number of solutions =4

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