When applying gradient descent to minimize a scalar quadratic function $f(x) = \frac{\lambda}{2} x^2$, the step-by-step update rule simplifies to $x_{t+1} = x_t - \eta \lambda x_t = (1 - \eta \lambda) x_t$, where $\eta$ is the learning rate and $\lambda$ represents the curvature. After $t$ iterations, the position is explicitly given by $x_t = (1 - \eta \lambda)^t x_0$. This demonstrates that the optimization converges exponentially toward the minimum at $x=0$ provided that the condition $|1 - \eta \lambda| < 1$ is met. This inequality shows that the convergence rate improves as $\eta$ increases until $\eta \lambda = 1$, but if the learning rate is too large such that $\eta \lambda > 2$, the sequence diverges entirely.