z^2-81z+196=0

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Solution for z^2-81z+196=0 equation:


Simplifying
z2 + -81z + 196 = 0

Reorder the terms:
196 + -81z + z2 = 0

Solving
196 + -81z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-196' to each side of the equation.
196 + -81z + -196 + z2 = 0 + -196

Reorder the terms:
196 + -196 + -81z + z2 = 0 + -196

Combine like terms: 196 + -196 = 0
0 + -81z + z2 = 0 + -196
-81z + z2 = 0 + -196

Combine like terms: 0 + -196 = -196
-81z + z2 = -196

The z term is -81z.  Take half its coefficient (-40.5).
Square it (1640.25) and add it to both sides.

Add '1640.25' to each side of the equation.
-81z + 1640.25 + z2 = -196 + 1640.25

Reorder the terms:
1640.25 + -81z + z2 = -196 + 1640.25

Combine like terms: -196 + 1640.25 = 1444.25
1640.25 + -81z + z2 = 1444.25

Factor a perfect square on the left side:
(z + -40.5)(z + -40.5) = 1444.25

Calculate the square root of the right side: 38.003289331

Break this problem into two subproblems by setting 
(z + -40.5) equal to 38.003289331 and -38.003289331.

Subproblem 1

z + -40.5 = 38.003289331 Simplifying z + -40.5 = 38.003289331 Reorder the terms: -40.5 + z = 38.003289331 Solving -40.5 + z = 38.003289331 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '40.5' to each side of the equation. -40.5 + 40.5 + z = 38.003289331 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + z = 38.003289331 + 40.5 z = 38.003289331 + 40.5 Combine like terms: 38.003289331 + 40.5 = 78.503289331 z = 78.503289331 Simplifying z = 78.503289331

Subproblem 2

z + -40.5 = -38.003289331 Simplifying z + -40.5 = -38.003289331 Reorder the terms: -40.5 + z = -38.003289331 Solving -40.5 + z = -38.003289331 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '40.5' to each side of the equation. -40.5 + 40.5 + z = -38.003289331 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + z = -38.003289331 + 40.5 z = -38.003289331 + 40.5 Combine like terms: -38.003289331 + 40.5 = 2.496710669 z = 2.496710669 Simplifying z = 2.496710669

Solution

The solution to the problem is based on the solutions from the subproblems. z = {78.503289331, 2.496710669}

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