3z^2-12z+9=1296

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Solution for 3z^2-12z+9=1296 equation: 


Simplifying
3z2 + -12z + 9 = 1296

Reorder the terms:
9 + -12z + 3z2 = 1296

Solving
9 + -12z + 3z2 = 1296

Solving for variable 'z'.

Reorder the terms:
9 + -1296 + -12z + 3z2 = 1296 + -1296

Combine like terms: 9 + -1296 = -1287
-1287 + -12z + 3z2 = 1296 + -1296

Combine like terms: 1296 + -1296 = 0
-1287 + -12z + 3z2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-429 + -4z + z2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-429 + -4z + z2)' equal to zero and attempt to solve:

Simplifying
-429 + -4z + z2 = 0

Solving
-429 + -4z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '429' to each side of the equation.
-429 + -4z + 429 + z2 = 0 + 429

Reorder the terms:
-429 + 429 + -4z + z2 = 0 + 429

Combine like terms: -429 + 429 = 0
0 + -4z + z2 = 0 + 429
-4z + z2 = 0 + 429

Combine like terms: 0 + 429 = 429
-4z + z2 = 429

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = 429 + 4

Reorder the terms:
4 + -4z + z2 = 429 + 4

Combine like terms: 429 + 4 = 433
4 + -4z + z2 = 433

Factor a perfect square on the left side:
(z + -2)(z + -2) = 433

Calculate the square root of the right side: 20.808652047

Break this problem into two subproblems by setting 
(z + -2) equal to 20.808652047 and -20.808652047.


Subproblem 1
z + -2 = 20.808652047

Simplifying
z + -2 = 20.808652047

Reorder the terms:
-2 + z = 20.808652047

Solving
-2 + z = 20.808652047

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = 20.808652047 + 2

Combine like terms: -2 + 2 = 0
0 + z = 20.808652047 + 2
z = 20.808652047 + 2

Combine like terms: 20.808652047 + 2 = 22.808652047
z = 22.808652047

Simplifying
z = 22.808652047


Subproblem 2
z + -2 = -20.808652047

Simplifying
z + -2 = -20.808652047

Reorder the terms:
-2 + z = -20.808652047

Solving
-2 + z = -20.808652047

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = -20.808652047 + 2

Combine like terms: -2 + 2 = 0
0 + z = -20.808652047 + 2
z = -20.808652047 + 2

Combine like terms: -20.808652047 + 2 = -18.808652047
z = -18.808652047

Simplifying
z = -18.808652047


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {22.808652047, -18.808652047}
Solution
z = {22.808652047, -18.808652047}

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