how do you solve 5/3 = 36t / (t^2 + 2)^2

Question

I can't figure out how to solve this question!!

Answer 1

5/3=36t/(t²+2)²,

5(t²+2)²=108t by cross-multiplying,

5(t⁴+4t²+4)=108t,

5t⁴+20t²-108t+20=0.

Let f(t)=5t⁴+20t²-108t+20, f'(t)=20t³+40t-108.

Newton's iterative Method is:

t_n+1=t_n-f(t_n)/f'(t_n) which creates t values which should converge to a root of the equation.

We need to choose an arbitrary value for t₀ to start the progression. Let's start with t₀=0, then:

t₁=-f(0)/f'(0)=-20/-108=5/27, t₂=0.1920706409, t₃=0.1920806052, t₄=0.1920806053, t₅≅t₄,

therefore one root is t=0.1920806053.

A graph shows that there's another root near t=2 so let t₀=2, then:

t₁=25/11, t₂=2.22103887, t₃=2.218732551, t₄=2.218728068, t₅≅t₄,

therefore another root is t=2.218728068.

By using synthetic division it's possible to find two more roots which, in this case, happen to be complex. The division will provide a quadratic which can be solved using the quadratic formula. However, for this question, we'll just keep to the real roots: t=0.1920806053 or 2.218728068 approximately. These numbers satisfy the given equation 5/3=36t/(t²+2)².