97z^2-2z-1=0

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Solution for 97z^2-2z-1=0 equation:


Simplifying
97z2 + -2z + -1 = 0

Reorder the terms:
-1 + -2z + 97z2 = 0

Solving
-1 + -2z + 97z2 = 0

Solving for variable 'z'.

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

Divide each side by '97'.
-0.01030927835 + -0.0206185567z + z2 = 0

Move the constant term to the right:

Add '0.01030927835' to each side of the equation.
-0.01030927835 + -0.0206185567z + 0.01030927835 + z2 = 0 + 0.01030927835

Reorder the terms:
-0.01030927835 + 0.01030927835 + -0.0206185567z + z2 = 0 + 0.01030927835

Combine like terms: -0.01030927835 + 0.01030927835 = 0.00000000000
0.00000000000 + -0.0206185567z + z2 = 0 + 0.01030927835
-0.0206185567z + z2 = 0 + 0.01030927835

Combine like terms: 0 + 0.01030927835 = 0.01030927835
-0.0206185567z + z2 = 0.01030927835

The z term is -0.0206185567z.  Take half its coefficient (-0.01030927835).
Square it (0.0001062812201) and add it to both sides.

Add '0.0001062812201' to each side of the equation.
-0.0206185567z + 0.0001062812201 + z2 = 0.01030927835 + 0.0001062812201

Reorder the terms:
0.0001062812201 + -0.0206185567z + z2 = 0.01030927835 + 0.0001062812201

Combine like terms: 0.01030927835 + 0.0001062812201 = 0.0104155595701
0.0001062812201 + -0.0206185567z + z2 = 0.0104155595701

Factor a perfect square on the left side:
(z + -0.01030927835)(z + -0.01030927835) = 0.0104155595701

Calculate the square root of the right side: 0.102056649

Break this problem into two subproblems by setting 
(z + -0.01030927835) equal to 0.102056649 and -0.102056649.

Subproblem 1

z + -0.01030927835 = 0.102056649 Simplifying z + -0.01030927835 = 0.102056649 Reorder the terms: -0.01030927835 + z = 0.102056649 Solving -0.01030927835 + z = 0.102056649 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.01030927835' to each side of the equation. -0.01030927835 + 0.01030927835 + z = 0.102056649 + 0.01030927835 Combine like terms: -0.01030927835 + 0.01030927835 = 0.00000000000 0.00000000000 + z = 0.102056649 + 0.01030927835 z = 0.102056649 + 0.01030927835 Combine like terms: 0.102056649 + 0.01030927835 = 0.11236592735 z = 0.11236592735 Simplifying z = 0.11236592735

Subproblem 2

z + -0.01030927835 = -0.102056649 Simplifying z + -0.01030927835 = -0.102056649 Reorder the terms: -0.01030927835 + z = -0.102056649 Solving -0.01030927835 + z = -0.102056649 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.01030927835' to each side of the equation. -0.01030927835 + 0.01030927835 + z = -0.102056649 + 0.01030927835 Combine like terms: -0.01030927835 + 0.01030927835 = 0.00000000000 0.00000000000 + z = -0.102056649 + 0.01030927835 z = -0.102056649 + 0.01030927835 Combine like terms: -0.102056649 + 0.01030927835 = -0.09174737065 z = -0.09174737065 Simplifying z = -0.09174737065

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.11236592735, -0.09174737065}

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