z^2-5z+2=0

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Solution for z^2-5z+2=0 equation: 


Simplifying
z2 + -5z + 2 = 0

Reorder the terms:
2 + -5z + z2 = 0

Solving
2 + -5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -5z + -2 + z2 = 0 + -2

Reorder the terms:
2 + -2 + -5z + z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -5z + z2 = 0 + -2
-5z + z2 = 0 + -2

Combine like terms: 0 + -2 = -2
-5z + z2 = -2

The z term is -5z.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5z + 6.25 + z2 = -2 + 6.25

Reorder the terms:
6.25 + -5z + z2 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + -5z + z2 = 4.25

Factor a perfect square on the left side:
(z + -2.5)(z + -2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(z + -2.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
z + -2.5 = 2.061552813

Simplifying
z + -2.5 = 2.061552813

Reorder the terms:
-2.5 + z = 2.061552813

Solving
-2.5 + z = 2.061552813

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + z = 2.061552813 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = 2.061552813 + 2.5
z = 2.061552813 + 2.5

Combine like terms: 2.061552813 + 2.5 = 4.561552813
z = 4.561552813

Simplifying
z = 4.561552813


Subproblem 2
z + -2.5 = -2.061552813

Simplifying
z + -2.5 = -2.061552813

Reorder the terms:
-2.5 + z = -2.061552813

Solving
-2.5 + z = -2.061552813

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + z = -2.061552813 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = -2.061552813 + 2.5
z = -2.061552813 + 2.5

Combine like terms: -2.061552813 + 2.5 = 0.438447187
z = 0.438447187

Simplifying
z = 0.438447187


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.561552813, 0.438447187}

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