4.
We ended up with 1.41421
Iteration

X

1

5.1

2

2.746078431

3

1.737194874

4

1.444238095

5

1.41425655

6

1.414213597

7

1.414213562

8

1.414213562

5.
Iteration

X

1

1

2

1.5

3

1.4166666

4

1.414215686

5

1.414213562

6

1.414213562

6.
Iteration

X

1

2

2

1.5

3

1.4166666

4

1.414215686

5

1.414213562

6

1.414213562

7. it took about 6 iterations to get to the exact answer.
8.
You change the equation to 12(x+3x) in order to get 3and for other numbers, you can change the number of the fraction where the denominator is x. So for 7, it would be 7 over x with the same equation. So exactly you would only be changing the number of the numerator in the second fraction to the whatever number inside the square root you are trying to find.
Iteration

x

1

2

2

1.732142857

3

1.73205081

4

1.732050808

5

1.732050808

9.
k=12(x+kx)
k=(x+kx)2/4
k=(x2+2k+k2x2)/4
4k=x2+2k+k2x2
4x2k=x4+2x2k+k2
x42x2k+k2=0
(x2k)2=0
(x2k)2=0
x2k=0
k=x2
k=x
Now when you input root K into the equation, it will be k=12(k+kk) and when you get rid of the root in the denominator, it will be 12(k+kkk), then when you simplify it, it will be, 12(k+k), which will be 12(2k)meaning k=k which shows that it has to be exact because they are equal values that are input.
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