I am working on a quadratically constrained quadratic program (QCQP) of the form:$$ \min_{x} \quad \frac{1}{2} x^T P x + q^T x + r$$ $$ \text{subject to} \qquad x^{T}x \leq 1 $$

where $P \in S^{++}_{n}$ is a symmetric positive definite matrix, and $q \in \mathbb{R}^{n}$ is a given vector. I aim to show that the optimal solution $x^{*}$ satisfies:$$ x^{*}=-(P+ \lambda I)^{-1}q$$ where $ \lambda$ =$\max\{ 0,\bar{\lambda}\}$ and $\bar{\lambda}$ is the largest solution to the nonlinear equation:$$q^{T}(P+\lambda I)^{-2}q=1 $$ I tried solving this by using the necessary and sufficient conditions for the optimal solution:$$ x \text{ is optimal if and only if } x \in X \text{ and} \\ \nabla f_0(x)^T (y - x) \geq 0 \quad \text{for all} \quad y \in X. $$ But I wasn't able to make progress.Is there an alternative approach to solve this problem, or any hints on how to use the optimality conditions more effectively? Any suggestions would be greatly appreciated!