They used the intercept format of a quadratic curve, y = k (x - a) (x - b) where a and b represent the two x-intercepts of the curve

Ok, I should have written as y = a (x - b) (x - c)

Here, the two x-intercepts are 0 and 120, so we get y = ax (x - 120)

To find a, we need another point on the curve - and the maximum point seems to be a good clue for finding the value of a

If you recall in Sec 2,

(x - b) (x - c) = 0
x - b = 0, x - c = 0
x = b, x = c

Here we are reversing the order to get (x - b) (x - c), but do note that multiplying (x - b) (x - c) by a will not change the solution

a (x - b) (x - c) = 0 leads to x = b or x = c

This multiplier a is quite important because many possible quadratic curves can be drawn to have the same two x-intercepts

This is why the format of the curve is y = a (x - b) (x - c)

I see, thanks for the detailed explanation, in part (iii) though, why did the qn find the difference between the 2 x to get the width of the arch?

Misunderstood part iii just now, so I am updating.

When y = 20, there are two possible values of x. These two values represent the two possible positions where the bridge ceiling is exactly 20 metres above the ground.

They wish to find the width (horizontal distance”) between the two said points, so we take the difference between the two values.

In the context of the story, this would mean that only vehicles 20 m tall of width smaller than this difference will be able to fit the bridge. This applies especially to construction cranes.

I get it, thanks for the explanation!