7z^2-7z+1=0

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Solution for 7z^2-7z+1=0 equation:


Simplifying
7z2 + -7z + 1 = 0

Reorder the terms:
1 + -7z + 7z2 = 0

Solving
1 + -7z + 7z2 = 0

Solving for variable 'z'.

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

Divide each side by '7'.
0.1428571429 + -1z + z2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -1z + -0.1428571429 + z2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + -1z + z2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + -1z + z2 = 0 + -0.1428571429
-1z + z2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
-1z + z2 = -0.1428571429

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = -0.1428571429 + 0.25

Reorder the terms:
0.25 + -1z + z2 = -0.1428571429 + 0.25

Combine like terms: -0.1428571429 + 0.25 = 0.1071428571
0.25 + -1z + z2 = 0.1071428571

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 0.1071428571

Calculate the square root of the right side: 0.327326835

Break this problem into two subproblems by setting 
(z + -0.5) equal to 0.327326835 and -0.327326835.

Subproblem 1

z + -0.5 = 0.327326835 Simplifying z + -0.5 = 0.327326835 Reorder the terms: -0.5 + z = 0.327326835 Solving -0.5 + z = 0.327326835 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 0.327326835 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 0.327326835 + 0.5 z = 0.327326835 + 0.5 Combine like terms: 0.327326835 + 0.5 = 0.827326835 z = 0.827326835 Simplifying z = 0.827326835

Subproblem 2

z + -0.5 = -0.327326835 Simplifying z + -0.5 = -0.327326835 Reorder the terms: -0.5 + z = -0.327326835 Solving -0.5 + z = -0.327326835 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -0.327326835 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -0.327326835 + 0.5 z = -0.327326835 + 0.5 Combine like terms: -0.327326835 + 0.5 = 0.172673165 z = 0.172673165 Simplifying z = 0.172673165

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.827326835, 0.172673165}

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