Z^2-37Z+24=0

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Solution for Z^2-37Z+24=0 equation: 


Simplifying
Z2 + -37Z + 24 = 0

Reorder the terms:
24 + -37Z + Z2 = 0

Solving
24 + -37Z + Z2 = 0

Solving for variable 'Z'.

Begin completing the square.

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + -37Z + -24 + Z2 = 0 + -24

Reorder the terms:
24 + -24 + -37Z + Z2 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + -37Z + Z2 = 0 + -24
-37Z + Z2 = 0 + -24

Combine like terms: 0 + -24 = -24
-37Z + Z2 = -24

The Z term is -37Z.  Take half its coefficient (-18.5).
Square it (342.25) and add it to both sides.

Add '342.25' to each side of the equation.
-37Z + 342.25 + Z2 = -24 + 342.25

Reorder the terms:
342.25 + -37Z + Z2 = -24 + 342.25

Combine like terms: -24 + 342.25 = 318.25
342.25 + -37Z + Z2 = 318.25

Factor a perfect square on the left side:
(Z + -18.5)(Z + -18.5) = 318.25

Calculate the square root of the right side: 17.839562775

Break this problem into two subproblems by setting 
(Z + -18.5) equal to 17.839562775 and -17.839562775.


Subproblem 1
Z + -18.5 = 17.839562775

Simplifying
Z + -18.5 = 17.839562775

Reorder the terms:
-18.5 + Z = 17.839562775

Solving
-18.5 + Z = 17.839562775

Solving for variable 'Z'.

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

Add '18.5' to each side of the equation.
-18.5 + 18.5 + Z = 17.839562775 + 18.5

Combine like terms: -18.5 + 18.5 = 0.0
0.0 + Z = 17.839562775 + 18.5
Z = 17.839562775 + 18.5

Combine like terms: 17.839562775 + 18.5 = 36.339562775
Z = 36.339562775

Simplifying
Z = 36.339562775


Subproblem 2
Z + -18.5 = -17.839562775

Simplifying
Z + -18.5 = -17.839562775

Reorder the terms:
-18.5 + Z = -17.839562775

Solving
-18.5 + Z = -17.839562775

Solving for variable 'Z'.

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

Add '18.5' to each side of the equation.
-18.5 + 18.5 + Z = -17.839562775 + 18.5

Combine like terms: -18.5 + 18.5 = 0.0
0.0 + Z = -17.839562775 + 18.5
Z = -17.839562775 + 18.5

Combine like terms: -17.839562775 + 18.5 = 0.660437225
Z = 0.660437225

Simplifying
Z = 0.660437225


Solution
The solution to the problem is based on the solutions
from the subproblems.
Z = {36.339562775, 0.660437225}

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