A curve C, has equation $$y = x^2 - 4x + k,$$ where $$k$$ is a constant - AQA - A-Level Maths Mechanics - Question 6 - 2017 - Paper 2

To find the value of $k$, we start by noting that the curve crosses the x-axis at the given points. This means that at these points, the value of $y$ is zero.

Given the points are $(2 + ext{√}5)$ and $(2 - ext{√}5)$, we can substitute one of these values into the equation:

1. Substitute $x = 2 + ext{√}5$: $0 = (2 + ext{√}5)^2 - 4(2 + ext{√}5) + k$
2. Expand and simplify: $0 = (4 + 4 ext{√}5 + 5) - (8 + 4 ext{√}5) + k$ $0 = 9 - 8 + k$
3. Hence, we find: $k = -1$

Thus, the value of $k$ is $-1$.