z^2+z-1=0

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


Simplifying
z2 + z + -1 = 0

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

Solving
-1 + z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The z term is z.  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.
z + 0.25 + z2 = 1 + 0.25

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

Combine like terms: 1 + 0.25 = 1.25
0.25 + z + z2 = 1.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

z + 0.5 = 1.118033989 Simplifying z + 0.5 = 1.118033989 Reorder the terms: 0.5 + z = 1.118033989 Solving 0.5 + z = 1.118033989 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 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.118033989 + -0.5 z = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 z = 0.618033989 Simplifying z = 0.618033989

Subproblem 2

z + 0.5 = -1.118033989 Simplifying z + 0.5 = -1.118033989 Reorder the terms: 0.5 + z = -1.118033989 Solving 0.5 + z = -1.118033989 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 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.118033989 + -0.5 z = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 z = -1.618033989 Simplifying z = -1.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.618033989, -1.618033989}

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