7z^2-2z-1=0

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


Simplifying
7z2 + -2z + -1 = 0

Reorder the terms:
-1 + -2z + 7z2 = 0

Solving
-1 + -2z + 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 + -0.2857142857z + z2 = 0

Move the constant term to the right:

Add '0.1428571429' to each side of the equation.
-0.1428571429 + -0.2857142857z + 0.1428571429 + z2 = 0 + 0.1428571429

Reorder the terms:
-0.1428571429 + 0.1428571429 + -0.2857142857z + z2 = 0 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + -0.2857142857z + z2 = 0 + 0.1428571429
-0.2857142857z + z2 = 0 + 0.1428571429

Combine like terms: 0 + 0.1428571429 = 0.1428571429
-0.2857142857z + z2 = 0.1428571429

The z term is -0.2857142857z.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857z + 0.02040816328 + z2 = 0.1428571429 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857z + z2 = 0.1428571429 + 0.02040816328

Combine like terms: 0.1428571429 + 0.02040816328 = 0.16326530618
0.02040816328 + -0.2857142857z + z2 = 0.16326530618

Factor a perfect square on the left side:
(z + -0.1428571429)(z + -0.1428571429) = 0.16326530618

Calculate the square root of the right side: 0.404061018

Break this problem into two subproblems by setting 
(z + -0.1428571429) equal to 0.404061018 and -0.404061018.

Subproblem 1

z + -0.1428571429 = 0.404061018 Simplifying z + -0.1428571429 = 0.404061018 Reorder the terms: -0.1428571429 + z = 0.404061018 Solving -0.1428571429 + z = 0.404061018 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + z = 0.404061018 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + z = 0.404061018 + 0.1428571429 z = 0.404061018 + 0.1428571429 Combine like terms: 0.404061018 + 0.1428571429 = 0.5469181609 z = 0.5469181609 Simplifying z = 0.5469181609

Subproblem 2

z + -0.1428571429 = -0.404061018 Simplifying z + -0.1428571429 = -0.404061018 Reorder the terms: -0.1428571429 + z = -0.404061018 Solving -0.1428571429 + z = -0.404061018 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + z = -0.404061018 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + z = -0.404061018 + 0.1428571429 z = -0.404061018 + 0.1428571429 Combine like terms: -0.404061018 + 0.1428571429 = -0.2612038751 z = -0.2612038751 Simplifying z = -0.2612038751

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.5469181609, -0.2612038751}

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