v^2-v-1=0

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Solution for v^2-v-1=0 equation:


Simplifying
v2 + -1v + -1 = 0

Reorder the terms:
-1 + -1v + v2 = 0

Solving
-1 + -1v + v2 = 0

Solving for variable 'v'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1v + 1 + v2 = 0 + 1

Reorder the terms:
-1 + 1 + -1v + v2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1v + v2 = 0 + 1
-1v + v2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1v + v2 = 1

The v term is -1v.  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.
-1v + 0.25 + v2 = 1 + 0.25

Reorder the terms:
0.25 + -1v + v2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1v + v2 = 1.25

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

Calculate the square root of the right side: 1.118033989

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

Subproblem 1

v + -0.5 = 1.118033989 Simplifying v + -0.5 = 1.118033989 Reorder the terms: -0.5 + v = 1.118033989 Solving -0.5 + v = 1.118033989 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + v = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + v = 1.118033989 + 0.5 v = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 v = 1.618033989 Simplifying v = 1.618033989

Subproblem 2

v + -0.5 = -1.118033989 Simplifying v + -0.5 = -1.118033989 Reorder the terms: -0.5 + v = -1.118033989 Solving -0.5 + v = -1.118033989 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + v = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + v = -1.118033989 + 0.5 v = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 v = -0.618033989 Simplifying v = -0.618033989

Solution

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

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