My functions are
y = -0.65(x -8.165)^2 + 1.5(x -8.165) + 6.872
y = 0.08(x-11)^3 -2.2(x-11) + 5.9
By using simultaneous equations and equating the functions to one another I've simplified it to the point where:
0.08x^3 - 1.99x^2 + 14.7255x - 27.608 = 0
This had me stuck for a while but when I looked it up online I found I could use the Newton - Raphson Method to solve it.
I got the intercepts: x = 2.85281 , x = 10.98682 , x = 11.03536
And although this helped me greatly, for the assignment I'm doing I won't be marked on using that method as we haven't been taught it and it's not on the criteria.
I'm just wondering if there is another method I can use that would get the same results.
Any help would be massively appreciated.
You know that you have three real roots.
Using whole numbers, your cubic equation write $$\frac{2 }{25}x^3-\frac{199 }{100}x^2+\frac{29451}{2000}x-\frac{22136643}{800000}=0$$
Use the trigonometric method as described in the Wikipedia page and get the nice $$x_k=\frac{199}{24}+\frac{19}{12} \sqrt{\frac{59}{5}} \cos \left(\frac{2 k\pi }{3}-\frac{1}{3} \cos ^{-1}\left(-\frac{69497939}{4046810 \sqrt{295}}\right)\right)$$ with $k=0,1,2$.