The number of real roots of the equation
`5+|2^(x)-1|=2^(x)(2^(x)-2)is`

Options

A. 1
B. 3
C. 4
D. 2

Answer

A

Explanation

Given equation `5+|2^(x)-1|=2^(x)(2^(x)-2)`
Case I
If `2^(x)-1 ge0impliesx ge0,then 5+2^(x)-1=2^(x)(2^(x)-2)`
Put `2^(x)=t,`then
`5+t-1=t^(2)-2timpliest^(2)-3t-4=0`
`impliest^(2)-4t+t-4=0impliest(t-4)+1(t-4)=0`
`impliest=4or-1impliest=4" "(becauset=2^(x)gt0)`
`implies2^(x)=4impliesx=2gt0`
`impliesx=2` is the solution.
m Case II
`If 2^(x)-1lt0impliesxlt0,`
then `5+1-2^(x)=2^(x)(2^(x)-2)`
Put `2^(x)=y, then 6-y=y^(2)-2y`
`impliesy^(2)-y-6=0impliesy^(2)-3y+2y-6=0`
`implies(y+2)(t-3)=0impliesy=3or -2`
`impliesy=3(as y=2^(x)gt0)implies2^(x)=3`
`impliesx=log_(2)3gt0`
So, `x=log_(2)3` is not a solution.
Therefore number of real roots is one.