2z^2-27z+14=0

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Solution for 2z^2-27z+14=0 equation: 


Simplifying
2z2 + -27z + 14 = 0

Reorder the terms:
14 + -27z + 2z2 = 0

Solving
14 + -27z + 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'.
7 + -13.5z + z2 = 0

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -13.5z + -7 + z2 = 0 + -7

Reorder the terms:
7 + -7 + -13.5z + z2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -13.5z + z2 = 0 + -7
-13.5z + z2 = 0 + -7

Combine like terms: 0 + -7 = -7
-13.5z + z2 = -7

The z term is -13.5z.  Take half its coefficient (-6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
-13.5z + 45.5625 + z2 = -7 + 45.5625

Reorder the terms:
45.5625 + -13.5z + z2 = -7 + 45.5625

Combine like terms: -7 + 45.5625 = 38.5625
45.5625 + -13.5z + z2 = 38.5625

Factor a perfect square on the left side:
(z + -6.75)(z + -6.75) = 38.5625

Calculate the square root of the right side: 6.209871174

Break this problem into two subproblems by setting 
(z + -6.75) equal to 6.209871174 and -6.209871174.


Subproblem 1
z + -6.75 = 6.209871174

Simplifying
z + -6.75 = 6.209871174

Reorder the terms:
-6.75 + z = 6.209871174

Solving
-6.75 + z = 6.209871174

Solving for variable 'z'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + z = 6.209871174 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + z = 6.209871174 + 6.75
z = 6.209871174 + 6.75

Combine like terms: 6.209871174 + 6.75 = 12.959871174
z = 12.959871174

Simplifying
z = 12.959871174


Subproblem 2
z + -6.75 = -6.209871174

Simplifying
z + -6.75 = -6.209871174

Reorder the terms:
-6.75 + z = -6.209871174

Solving
-6.75 + z = -6.209871174

Solving for variable 'z'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + z = -6.209871174 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + z = -6.209871174 + 6.75
z = -6.209871174 + 6.75

Combine like terms: -6.209871174 + 6.75 = 0.540128826
z = 0.540128826

Simplifying
z = 0.540128826


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {12.959871174, 0.540128826}

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