Why roundoff error in solving cubic equation?

Questions
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1

Hey guys!!

while solving the following cubic equation in Wolfram Mathematica 9

-0.363700352e-2*x^3-.4041941000*x^2+3.397775673*x-2.377540486 = 0

I got these three roots

x = .7709248124
x= 7.123944371
x= -119.0286907

But,

While looking for step to step solution, it shows round-off error at each roots. Why is it so?
I want answers in simple forum directly only by using these formulas. As i have to show the proof!

Both files are attached.

Thanks in advance,

2

You can avoid round-off error in the steps by rationalizing the problem and solving it symbolically and then finding the numerical value of the answer.

In[18]:= res = 
  Solve[Rationalize[(-0.363700352*10^-2)*x^3 - .4041941000*x^2 + 
      3.397775673*x - 2.377540486 == 0, 10^-16], x];

In[19]:= N[res, 16]

Out[19]= {{x -> 
   7.123944370913387 + 0.*10^-16 I}, {x -> -119.02869068703848 + 
    0.*10^-15 I}, {x -> 0.7709248124293782 + 0.*10^-17 I}}