Find the coordinates of the common points of the graph of the function y = x ^ 2-4x + | 2x-8 | and the x-axis.

1. Let’s transform the function:

y = x ^ 2 – 4x + | 2x – 8 |;
y = x (x – 4) + 2 | x – 4 |.
2. To find the abscissa of the intersection points of the function graph with the OX axis, we solve the equation:

x (x – 4) + 2 | x – 4 | = 0.

Consider two cases:

a) x ∈ (-∞; 4);

x (x – 4) – 2 (x – 4) = 0;
(x – 4) (x – 2) = 0;
[x – 4 = 0;
[x – 2 = 0;
[x = 4 ∉ (-∞; 4);
[x = 2 ∈ (-∞; 4).
b) x ∈ [4; ∞);

x (x – 4) + 2 (x – 4) = 0;
(x – 4) (x + 2) = 0;
[x – 4 = 0;
[x + 2 = 0;
[x = 4 ∈ [4; ∞);
[x = -2 ∉ [4; ∞).
3. Coordinates of points:

a) x = 2; y = 0;
b) x = 4; y = 0.
Answer: (2; 0); (4; 0).