2z^2-31z-8=0

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Solution for 2z^2-31z-8=0 equation: 


Simplifying
2z2 + -31z + -8 = 0

Reorder the terms:
-8 + -31z + 2z2 = 0

Solving
-8 + -31z + 2z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-4 + -15.5z + z2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -15.5z + 4 + z2 = 0 + 4

Reorder the terms:
-4 + 4 + -15.5z + z2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -15.5z + z2 = 0 + 4
-15.5z + z2 = 0 + 4

Combine like terms: 0 + 4 = 4
-15.5z + z2 = 4

The z term is -15.5z.  Take half its coefficient (-7.75).
Square it (60.0625) and add it to both sides.

Add '60.0625' to each side of the equation.
-15.5z + 60.0625 + z2 = 4 + 60.0625

Reorder the terms:
60.0625 + -15.5z + z2 = 4 + 60.0625

Combine like terms: 4 + 60.0625 = 64.0625
60.0625 + -15.5z + z2 = 64.0625

Factor a perfect square on the left side:
(z + -7.75)(z + -7.75) = 64.0625

Calculate the square root of the right side: 8.003905297

Break this problem into two subproblems by setting 
(z + -7.75) equal to 8.003905297 and -8.003905297.


Subproblem 1
z + -7.75 = 8.003905297

Simplifying
z + -7.75 = 8.003905297

Reorder the terms:
-7.75 + z = 8.003905297

Solving
-7.75 + z = 8.003905297

Solving for variable 'z'.

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

Add '7.75' to each side of the equation.
-7.75 + 7.75 + z = 8.003905297 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + z = 8.003905297 + 7.75
z = 8.003905297 + 7.75

Combine like terms: 8.003905297 + 7.75 = 15.753905297
z = 15.753905297

Simplifying
z = 15.753905297


Subproblem 2
z + -7.75 = -8.003905297

Simplifying
z + -7.75 = -8.003905297

Reorder the terms:
-7.75 + z = -8.003905297

Solving
-7.75 + z = -8.003905297

Solving for variable 'z'.

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

Add '7.75' to each side of the equation.
-7.75 + 7.75 + z = -8.003905297 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + z = -8.003905297 + 7.75
z = -8.003905297 + 7.75

Combine like terms: -8.003905297 + 7.75 = -0.253905297
z = -0.253905297

Simplifying
z = -0.253905297


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {15.753905297, -0.253905297}

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