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I need help with this question

 Feb 28, 2019
 #1
avatar+104899 
+1

We are trying to find the x that makes x^3 -3x - 1 = 0

 

We can use "divide and average" to find this

 

Le/t suppose that the answer of x = 1.85 =  0

 

So   (1.85)^3 -3(1.85) - 1  =   -.218

 

The root must be between  1.85 and 1.9

Average ths   [ 1.85 + 1.9] /2 = 1.875

Put this into the function

 (1.875)^3 -3(1.875) - 1   =  -.033                

 

The root must be between   1.875 and 1.9

Average these =  [ 1.875 + 1.9 ] / 2  = 1.8875

Put this value into the function

 (1.8875)^3 -3(1.8875) - 1   = .062

 

The root must be between 1.875 and 1.8875

Average these  = [ 1.8875 + 1.875 ] / 2 = 1.88125

Put thiis into the function

 (1.88125)^3 -3(1.88125) - 1  = .014

 

The root must be between 1.875 and 1.88125

Average these = [ 1.88125 + 1.875 ] / 2  = 1.878125

Put this into the function ≈  -.009       (1) 

 

The root must be between  1.878125 and 1.88125

Average these = 1.8796875

Put this into the function =  .002

 

The root must be between   1.878125 and 1.8796875  

Average these = 1.87890625

Put this into the function = -.0036     ......This agrees with (1) to 2 decimal places

 

So....the approximate root is  ≈  1.87

 

[ Note....the true root is ≈ 1.8794 ]

 

 

 

cool cool cool

 Feb 28, 2019
edited by CPhill  Feb 28, 2019
 #2
avatar+19832 
+1

Wellllll...since it is between 1.8 and 1.9   let's pick   1.85  and substitute that into the equation and look at the result to see if it = 1

 

1.85      1.85^3 - 3(1.85) = .7816     Kinda close, lets try  1.86

1.86                                =  .8548   closer...try 1.87

1.87                                =  .9292   getting there ....try 1.88

1.88                                = 1.0046     BINGO !   

 

1.88   to 2 decimal places  is a solution to  x^3 - 3x = 1

 Feb 28, 2019

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