Find number of solutions of the equation `sqrt((x+8)+2sqrt(x+7))+sqrt((x+1)-sqrt(x+7))=4`

Answer

`x_(1)=2`

Explanation

We have `sqrt(x+8+2sqrt((x+7)))+sqrt((x+1)-sqrt((x+7)))=4`..i
Let `sqrt((x+7))=lamda`..ii
Or `x=lamda^(2)-7`
Then Eq (i) reduces to
`sqrt((lamda^(2)-8+8+2lamdas))+sqrt((lamda^(2)-7+1-lamda))=4`
`implies(lamda+1)+sqrt((lamda^(2)-lamda-6))=4`
or `sqrt((lamda^(2)-lamda-6))=3-lamda`
On squaring both sides we get
`lamda^(2)-lamda-6=9+lamda^(2)-6lamda`
`implies5lamda=15`
`:.lamda=3`
`impliessqrt((x+7))=3`[from Eq. (ii) ]
or `x+7=9`
`:.x=2`
and `x=2` satisfies Eq. (i)
Hence `x_(1)=2`