If $A$ is a symmetric matrix, then can we write $\lambda_{min}(A)\|x\|^2 \leq x^TAx \leq \lambda_{max}(A)\|x\|^2$?

Question

I have a question and that is, if lets say we have a symmetric matrix $A$ and a vector $x$ of appropriate dimension, then can we have, the famous identity, \begin{equation} \lambda_{min}(A)\|x\|^2 \leq x^TAx \leq \lambda_{max}(A)\|x\|^2 \end{equation} My main concern is (please correct me if my reasoning is wrong!), this identity holds only when $A$ is diagonalizable or in other words when sum of all the eigenspace is equal to the size of the matrix. Otherwise, will this identity not become something like this? \begin{equation} \lambda_{min}(A)\|x\|^2 + \epsilon \leq x^TAx \leq \lambda_{max}(A)\|x\|^2+\epsilon \end{equation} The term $\epsilon$ depicting the change due the norm of extra terms that pop up due to the jordan form. Is my reasoning correct? If not, kindly help me find a bound over $x^TAx$ when $A$ happens to be just symmetric.