You're thinking too hard.

Sorry, but I'm going to change the variables s & t to x & y.values of s along y axis

values of t along x axis

s=f(t)=a (t-p)(t-q)^2

where a, p, q are constants

the curve meets the coordinate axes at the points (0, -4.5) (1, 0) (3, 0)

find values for a, p ,q

The 2nd and 3rd points are x-intercepts.

(Note: if you have an equation in factored form, like y = (x - m)(x - n), finding the x-intercepts is really easy; they're the numbers subtracted from x. In my example, they would be m and n.)

That means that p = 1 and q = 3,orp = 3 and q = 1. Plug in each pair, with the 1st point, to find a.

p = 1, q = 3:

p = 3, q = 1:

So I found two sets of values of a, p, and q. Unless there is more information that I'm missing...?

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