4455z-345z^2=678z^2-456

Solution for 4455z-345z^2=678z^2-456 equation:

Simplifying
4455z + -345z2 = 678z2 + -456

Reorder the terms:
4455z + -345z2 = -456 + 678z2

Solving
4455z + -345z2 = -456 + 678z2

Solving for variable 'z'.

Reorder the terms:
456 + 4455z + -345z2 + -678z2 = -456 + 678z2 + 456 + -678z2

Combine like terms: -345z2 + -678z2 = -1023z2
456 + 4455z + -1023z2 = -456 + 678z2 + 456 + -678z2

Reorder the terms:
456 + 4455z + -1023z2 = -456 + 456 + 678z2 + -678z2

Combine like terms: -456 + 456 = 0
456 + 4455z + -1023z2 = 0 + 678z2 + -678z2
456 + 4455z + -1023z2 = 678z2 + -678z2

Combine like terms: 678z2 + -678z2 = 0
456 + 4455z + -1023z2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(152 + 1485z + -341z2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(152 + 1485z + -341z2)' equal to zero and attempt to solve:

Simplifying
152 + 1485z + -341z2 = 0

Solving
152 + 1485z + -341z2 = 0

Begin completing the square.  Divide all terms by
-341 the coefficient of the squared term: 

Divide each side by '-341'.
-0.4457478006 + -4.35483871z + z2 = 0

Move the constant term to the right:

Add '0.4457478006' to each side of the equation.
-0.4457478006 + -4.35483871z + 0.4457478006 + z2 = 0 + 0.4457478006

Reorder the terms:
-0.4457478006 + 0.4457478006 + -4.35483871z + z2 = 0 + 0.4457478006

Combine like terms: -0.4457478006 + 0.4457478006 = 0.0000000000
0.0000000000 + -4.35483871z + z2 = 0 + 0.4457478006
-4.35483871z + z2 = 0 + 0.4457478006

Combine like terms: 0 + 0.4457478006 = 0.4457478006
-4.35483871z + z2 = 0.4457478006

The z term is -4.35483871z.  Take half its coefficient (-2.177419355).
Square it (4.741155048) and add it to both sides.

Add '4.741155048' to each side of the equation.
-4.35483871z + 4.741155048 + z2 = 0.4457478006 + 4.741155048

Reorder the terms:
4.741155048 + -4.35483871z + z2 = 0.4457478006 + 4.741155048

Combine like terms: 0.4457478006 + 4.741155048 = 5.1869028486
4.741155048 + -4.35483871z + z2 = 5.1869028486

Factor a perfect square on the left side:
(z + -2.177419355)(z + -2.177419355) = 5.1869028486

Calculate the square root of the right side: 2.277477299

Break this problem into two subproblems by setting 
(z + -2.177419355) equal to 2.277477299 and -2.277477299.

Subproblem 1
z + -2.177419355 = 2.277477299

Simplifying
z + -2.177419355 = 2.277477299

Reorder the terms:
-2.177419355 + z = 2.277477299

Solving
-2.177419355 + z = 2.277477299

Solving for variable 'z'.

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

Add '2.177419355' to each side of the equation.
-2.177419355 + 2.177419355 + z = 2.277477299 + 2.177419355

Combine like terms: -2.177419355 + 2.177419355 = 0.000000000
0.000000000 + z = 2.277477299 + 2.177419355
z = 2.277477299 + 2.177419355

Combine like terms: 2.277477299 + 2.177419355 = 4.454896654
z = 4.454896654

Simplifying
z = 4.454896654

Subproblem 2
z + -2.177419355 = -2.277477299

Simplifying
z + -2.177419355 = -2.277477299

Reorder the terms:
-2.177419355 + z = -2.277477299

Solving
-2.177419355 + z = -2.277477299

Solving for variable 'z'.

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

Add '2.177419355' to each side of the equation.
-2.177419355 + 2.177419355 + z = -2.277477299 + 2.177419355

Combine like terms: -2.177419355 + 2.177419355 = 0.000000000
0.000000000 + z = -2.277477299 + 2.177419355
z = -2.277477299 + 2.177419355

Combine like terms: -2.277477299 + 2.177419355 = -0.100057944
z = -0.100057944

Simplifying
z = -0.100057944

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.454896654, -0.100057944}
Solution
z = {4.454896654, -0.100057944}