z^2-z-162=0

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Solution for z^2-z-162=0 equation:


Simplifying
z2 + -1z + -162 = 0

Reorder the terms:
-162 + -1z + z2 = 0

Solving
-162 + -1z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '162' to each side of the equation.
-162 + -1z + 162 + z2 = 0 + 162

Reorder the terms:
-162 + 162 + -1z + z2 = 0 + 162

Combine like terms: -162 + 162 = 0
0 + -1z + z2 = 0 + 162
-1z + z2 = 0 + 162

Combine like terms: 0 + 162 = 162
-1z + z2 = 162

The z term is -1z.  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.
-1z + 0.25 + z2 = 162 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 162 + 0.25

Combine like terms: 162 + 0.25 = 162.25
0.25 + -1z + z2 = 162.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 162.25

Calculate the square root of the right side: 12.737739203

Break this problem into two subproblems by setting 
(z + -0.5) equal to 12.737739203 and -12.737739203.

Subproblem 1

z + -0.5 = 12.737739203 Simplifying z + -0.5 = 12.737739203 Reorder the terms: -0.5 + z = 12.737739203 Solving -0.5 + z = 12.737739203 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 12.737739203 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 12.737739203 + 0.5 z = 12.737739203 + 0.5 Combine like terms: 12.737739203 + 0.5 = 13.237739203 z = 13.237739203 Simplifying z = 13.237739203

Subproblem 2

z + -0.5 = -12.737739203 Simplifying z + -0.5 = -12.737739203 Reorder the terms: -0.5 + z = -12.737739203 Solving -0.5 + z = -12.737739203 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -12.737739203 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -12.737739203 + 0.5 z = -12.737739203 + 0.5 Combine like terms: -12.737739203 + 0.5 = -12.237739203 z = -12.237739203 Simplifying z = -12.237739203

Solution

The solution to the problem is based on the solutions from the subproblems. z = {13.237739203, -12.237739203}

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