Using $x_{t} = -2 - \frac{4}{x_{t}}$ with $x_{t} = -2.5$ (a) find the values of $x_{t}$, $x_{1}$, and $x_{2}$ - Edexcel - GCSE Maths - Question 16 - 2017 - Paper 3

The values of $x_{1}$, $x_{2}$, and $x_{t}$ are related through the iterative method of solving the equation. In particular, they represent successive approximations of a solution to the quadratic equation $x^{2} + 2x + 4 = 0$. The equation has complex solutions given that its discriminant ($b^2 - 4ac = 2^2 - 4(1)(4) < 0$) is negative.

Thus, $x_{1}$ and subsequent iterations are estimates that converge to one of these complex solutions. The iterative process helps in estimating solutions for nonlinear equations and illustrates how different $x$ values lead back to the original equation.