From Förberedande kurs i matematik 1

We rewrite the expression by completing the square:

\displaystyle \begin{align} & -x^{2}+3x-4=-\left( x^{2}-3x+4 \right)=-\left( \left( x-\frac{3}{2} \right)^{2}-\left( \frac{3}{2} \right)^{2}+4 \right) \\ & =-\left( \left( x-\frac{3}{2} \right)^{2}-\frac{9}{4}+\frac{16}{4} \right)=-\left( \left( x-\frac{3}{2} \right)^{2}+\frac{7}{4} \right)=-\left( x-\frac{3}{2} \right)^{2}-\frac{7}{4} \\ \end{align}

Now, we see that the first term \displaystyle -\left( x-\frac{3}{2} \right)^{2} is a quadratic with a minus sign in front, so that term is always less than or equal to zero. This means that the polynomial's largest value is \displaystyle -{7}/{4}\; and that occurs when \displaystyle x-\frac{3}{2}=0, i.e. \displaystyle x=\frac{3}{2}.