546000t=8t^2+8t+450

Solution for 546000t=8t^2+8t+450 equation:

Simplifying
546000t = 8t2 + 8t + 450

Reorder the terms:
546000t = 450 + 8t + 8t2

Solving
546000t = 450 + 8t + 8t2

Solving for variable 't'.

Reorder the terms:
-450 + 546000t + -8t + -8t2 = 450 + 8t + 8t2 + -450 + -8t + -8t2

Combine like terms: 546000t + -8t = 545992t
-450 + 545992t + -8t2 = 450 + 8t + 8t2 + -450 + -8t + -8t2

Reorder the terms:
-450 + 545992t + -8t2 = 450 + -450 + 8t + -8t + 8t2 + -8t2

Combine like terms: 450 + -450 = 0
-450 + 545992t + -8t2 = 0 + 8t + -8t + 8t2 + -8t2
-450 + 545992t + -8t2 = 8t + -8t + 8t2 + -8t2

Combine like terms: 8t + -8t = 0
-450 + 545992t + -8t2 = 0 + 8t2 + -8t2
-450 + 545992t + -8t2 = 8t2 + -8t2

Combine like terms: 8t2 + -8t2 = 0
-450 + 545992t + -8t2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-225 + 272996t + -4t2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-225 + 272996t + -4t2)' equal to zero and attempt to solve:

Simplifying
-225 + 272996t + -4t2 = 0

Solving
-225 + 272996t + -4t2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
56.25 + -68249t + t2 = 0

Move the constant term to the right:

Add '-56.25' to each side of the equation.
56.25 + -68249t + -56.25 + t2 = 0 + -56.25

Reorder the terms:
56.25 + -56.25 + -68249t + t2 = 0 + -56.25

Combine like terms: 56.25 + -56.25 = 0.00
0.00 + -68249t + t2 = 0 + -56.25
-68249t + t2 = 0 + -56.25

Combine like terms: 0 + -56.25 = -56.25
-68249t + t2 = -56.25

The t term is -68249t.  Take half its coefficient (-34124.5).
Square it (1164481500) and add it to both sides.

Add '1164481500' to each side of the equation.
-68249t + 1164481500 + t2 = -56.25 + 1164481500

Reorder the terms:
1164481500 + -68249t + t2 = -56.25 + 1164481500

Combine like terms: -56.25 + 1164481500 = 1164481443.75
1164481500 + -68249t + t2 = 1164481443.75

Factor a perfect square on the left side:
(t + -34124.5)(t + -34124.5) = 1164481443.75

Calculate the square root of the right side: 34124.499172149

Break this problem into two subproblems by setting 
(t + -34124.5) equal to 34124.499172149 and -34124.499172149.

Subproblem 1
t + -34124.5 = 34124.499172149

Simplifying
t + -34124.5 = 34124.499172149

Reorder the terms:
-34124.5 + t = 34124.499172149

Solving
-34124.5 + t = 34124.499172149

Solving for variable 't'.

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

Add '34124.5' to each side of the equation.
-34124.5 + 34124.5 + t = 34124.499172149 + 34124.5

Combine like terms: -34124.5 + 34124.5 = 0.0
0.0 + t = 34124.499172149 + 34124.5
t = 34124.499172149 + 34124.5

Combine like terms: 34124.499172149 + 34124.5 = 68248.999172149
t = 68248.999172149

Simplifying
t = 68248.999172149

Subproblem 2
t + -34124.5 = -34124.499172149

Simplifying
t + -34124.5 = -34124.499172149

Reorder the terms:
-34124.5 + t = -34124.499172149

Solving
-34124.5 + t = -34124.499172149

Solving for variable 't'.

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

Add '34124.5' to each side of the equation.
-34124.5 + 34124.5 + t = -34124.499172149 + 34124.5

Combine like terms: -34124.5 + 34124.5 = 0.0
0.0 + t = -34124.499172149 + 34124.5
t = -34124.499172149 + 34124.5

Combine like terms: -34124.499172149 + 34124.5 = 0.000827851
t = 0.000827851

Simplifying
t = 0.000827851

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {68248.999172149, 0.000827851}
Solution
t = {68248.999172149, 0.000827851}