z^2-3z+1=0

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Solution for z^2-3z+1=0 equation:


Simplifying
z2 + -3z + 1 = 0

Reorder the terms:
1 + -3z + z2 = 0

Solving
1 + -3z + 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 + -3z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + -3z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3z + z2 = 0 + -1
-3z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3z + z2 = -1

The z term is -3z.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3z + 2.25 + z2 = -1 + 2.25

Reorder the terms:
2.25 + -3z + z2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + -3z + z2 = 1.25

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z + -1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

z + -1.5 = 1.118033989 Simplifying z + -1.5 = 1.118033989 Reorder the terms: -1.5 + z = 1.118033989 Solving -1.5 + z = 1.118033989 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 1.118033989 + 1.5 z = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 z = 2.618033989 Simplifying z = 2.618033989

Subproblem 2

z + -1.5 = -1.118033989 Simplifying z + -1.5 = -1.118033989 Reorder the terms: -1.5 + z = -1.118033989 Solving -1.5 + z = -1.118033989 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -1.118033989 + 1.5 z = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 z = 0.381966011 Simplifying z = 0.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.618033989, 0.381966011}

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