This is likely to be a lengthy solution so I may need to abbreviate some of the details. But first we need some preliminaries. Here is a table and a graph. f(x) is shown in red. The blue curve is a parabola attempting to match the shape of the curve using three points near to a zero. The roots, or zeroes, are the 4 points of intersection of the red curve and the x-axis.
Now to find the parabola’s equation. There are various ways to do this but I’ll use simultaneous equations.
g₁(x)=ax²+bx+c. To find a, b, c using x₀=-15, x₁=-10, x₂=-5.
(1) 40.2926=225a-15b+c
(2) -35.6704=100a-10b+c
(3) -12.2184=25a-5b+c
(1)-(2): 75.963=125a-5b⇒-151.926=-250a+10b
(1)-(3): 52.511=200a-10b
Adding the last two equations: -99.415=-50a, a=1.9883.
From this b=34.5149, c=110.6486.
g(x)=1.9883x²+34.5149x+110.6486 (blue curve).
More to follow...
