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#1**+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 ]

CPhill Feb 28, 2019

#2**+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

ElectricPavlov Feb 28, 2019