Question is below regarding a quadratic equation in physics?

Question

Assuming that the relationship between spring compression and height is quadratic, how far will the ball go at the highest setting, which compresses the spring 3 ?

Answer 1

OP

h

------------------------- uncompressed level, x=0

--------------------------- A (x= -x) compressed by (x)

energy conservation> initial (at A) = final (at P)

0.5 kx^2 + mg (-x) = mgh

h = [k/2mg] x^2 - x -------- (1)

its quadratic

dh/dx = [h/mg] x - 1 = 0  for maximum

x (for max) = mg/k

given x = 3 = mg/k

mg = 3k >>> condition

h(max) = [k/(2*3k)] [-3]^2 - [-3]

h(max) = [9/6] + 3 = 1.5+3 = 4.5

h(max) = 4.5

from uncompressed level

from compressed level = 3+4.5 = 7.5

>> its tricky