Find the Values of \(x\) for Which \(f(x)=g(x)\)
Question
The functions
\[ f(x)=3x^2-1 \]and
\[ g(x)=3+x \]are equal for which values of \(x\)?
Solution
Given
\[ f(x)=3x^2-1 \]and
\[ g(x)=3+x \]To find the values of \(x\) for which the functions are equal, equate them:
\[ 3x^2-1=3+x \]Bring all terms to one side:
\[ 3x^2-x-4=0 \]Factorize:
\[ 3x^2-4x+3x-4=0 \] \[ x(3x-4)+1(3x-4)=0 \] \[ (x+1)(3x-4)=0 \]Therefore,
\[ x+1=0 \quad \text{or} \quad 3x-4=0 \] \[ x=-1 \quad \text{or} \quad x=\frac43 \]