Mathematics - Quadratic Equation Question with Solution | TestHub

The minimum value of a quadratic expression $a{x}^2+bx+c$ with $a>0$ occurs at $x=-b/(2a)$. For $f(x)=x^2-4x+7$, we have $a=1$ and $b=-4$. So, $x=-(-4)/(2\cdot 1)=4/2=2$. The minimum value is $f(2)=(2{)}^2-4(2)+7=4-8+7=3$. Alternatively, by completing the square, $f(x)=(x^2-4x+4)+3=(x-2{)}^2+3$. Since $(x-2{)}^2\ge 0$, the minimum value of $f(x)$ is $0+3=3$.