64z^2-144144z+1=0

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Solution for 64z^2-144144z+1=0 equation: 


Simplifying
64z2 + -144144z + 1 = 0

Reorder the terms:
1 + -144144z + 64z2 = 0

Solving
1 + -144144z + 64z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
64 the coefficient of the squared term: 

Divide each side by '64'.
0.015625 + -2252.25z + z2 = 0

Move the constant term to the right:

Add '-0.015625' to each side of the equation.
0.015625 + -2252.25z + -0.015625 + z2 = 0 + -0.015625

Reorder the terms:
0.015625 + -0.015625 + -2252.25z + z2 = 0 + -0.015625

Combine like terms: 0.015625 + -0.015625 = 0.000000
0.000000 + -2252.25z + z2 = 0 + -0.015625
-2252.25z + z2 = 0 + -0.015625

Combine like terms: 0 + -0.015625 = -0.015625
-2252.25z + z2 = -0.015625

The z term is -2252.25z.  Take half its coefficient (-1126.125).
Square it (1268157.516) and add it to both sides.

Add '1268157.516' to each side of the equation.
-2252.25z + 1268157.516 + z2 = -0.015625 + 1268157.516

Reorder the terms:
1268157.516 + -2252.25z + z2 = -0.015625 + 1268157.516

Combine like terms: -0.015625 + 1268157.516 = 1268157.500375
1268157.516 + -2252.25z + z2 = 1268157.500375

Factor a perfect square on the left side:
(z + -1126.125)(z + -1126.125) = 1268157.500375

Calculate the square root of the right side: 1126.124993229

Break this problem into two subproblems by setting 
(z + -1126.125) equal to 1126.124993229 and -1126.124993229.


Subproblem 1
z + -1126.125 = 1126.124993229

Simplifying
z + -1126.125 = 1126.124993229

Reorder the terms:
-1126.125 + z = 1126.124993229

Solving
-1126.125 + z = 1126.124993229

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '1126.125' to each side of the equation.
-1126.125 + 1126.125 + z = 1126.124993229 + 1126.125

Combine like terms: -1126.125 + 1126.125 = 0.000
0.000 + z = 1126.124993229 + 1126.125
z = 1126.124993229 + 1126.125

Combine like terms: 1126.124993229 + 1126.125 = 2252.249993229
z = 2252.249993229

Simplifying
z = 2252.249993229


Subproblem 2
z + -1126.125 = -1126.124993229

Simplifying
z + -1126.125 = -1126.124993229

Reorder the terms:
-1126.125 + z = -1126.124993229

Solving
-1126.125 + z = -1126.124993229

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '1126.125' to each side of the equation.
-1126.125 + 1126.125 + z = -1126.124993229 + 1126.125

Combine like terms: -1126.125 + 1126.125 = 0.000
0.000 + z = -1126.124993229 + 1126.125
z = -1126.124993229 + 1126.125

Combine like terms: -1126.124993229 + 1126.125 = 0.000006771
z = 0.000006771

Simplifying
z = 0.000006771


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2252.249993229, 0.000006771}

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