How would you find the coordinates of the points of intersection between a quadratic equation and a linear equation?

We have to be careful with the terminology here. A quadratic curve can intersect a line, but a quadratic equation doesn't intersect a linear equation.

Consider the quadratic function f(x) = ax^2 + bx + c and the linear function g(x) = px + q. We can plot both the quadratic curve y = f(x) and the line y = g(x) on a graph. If the curve and the line intersect, then we can find the x-coordinate(s) of the intersection by solving f(x) = g(x), i.e. the quadratic equation ax^2 + (b - p)x + (c - q) = 0. We then substitute the x-coordinate(s) into y = g(x) (or f(x)) to find the corresponding y-coordinate(s).

For example, if f(x) = 2x^2 - 4x + 3 and g(x) = 2x - 1, then the x-coordinates of the points of intersection of y = f(x) and y = g(x) are found by solving 2x^2 - 4x + 3 = 2x - 1, i.e. 2x^2 - 6x + 4 = 0. This factorises to 2(x - 1)(x - 2) = 0, so the x-coordinates are x = 1 and x = 2. Substituting into y = g(x) gives the corresponding y-coordinates y = 1 and y = 3. So the coordinates are (1, 1) and (2, 3).