√log₁₀x +log_10√x = - ½

This equation can be written as (log₁₀x)^1/2+log₁₀x^1/2=-1/2

or

(log₁₀x)^1/2+1/2log₁₀x=-1/2

Put (log₁₀x)=t²

Now

t+t²/2=-1/2

(t+1)²=0

t=-1

(log₁₀x)=1

x=10¹=10