Solve \(f(x)=0\) for \(f(x)=[x]^2-5[x]+6\)
Question
If
\[ f(x)=[x]^2-5[x]+6 \]then the set of values of \(x\) satisfying
\[ f(x)=0 \]is __________.
Solution
Given
\[ f(x)=[x]^2-5[x]+6 \]We need to solve
\[ [x]^2-5[x]+6=0 \]Let
\[ [x]=t \]Then,
\[ t^2-5t+6=0 \]Factorize:
\[ (t-2)(t-3)=0 \]Therefore,
\[ t=2 \quad \text{or} \quad t=3 \]Since \(t=[x]\),
\[ [x]=2 \quad \text{or} \quad [x]=3 \]Case 1: \([x]=2\)
\[ 2\le x<3 \]Case 2: \([x]=3\)
\[ 3\le x<4 \]Combining both intervals,
\[ 2\le x<4 \]