6(t)=t^2-3t+1

Solution for 6(t)=t^2-3t+1 equation:

Simplifying
6(t) = t2 + -3t + 1

Reorder the terms:
6t = 1 + -3t + t2

Solving
6t = 1 + -3t + t2

Solving for variable 't'.

Reorder the terms:
-1 + 6t + 3t + -1t2 = 1 + -3t + t2 + -1 + 3t + -1t2

Combine like terms: 6t + 3t = 9t
-1 + 9t + -1t2 = 1 + -3t + t2 + -1 + 3t + -1t2

Reorder the terms:
-1 + 9t + -1t2 = 1 + -1 + -3t + 3t + t2 + -1t2

Combine like terms: 1 + -1 = 0
-1 + 9t + -1t2 = 0 + -3t + 3t + t2 + -1t2
-1 + 9t + -1t2 = -3t + 3t + t2 + -1t2

Combine like terms: -3t + 3t = 0
-1 + 9t + -1t2 = 0 + t2 + -1t2
-1 + 9t + -1t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
-1 + 9t + -1t2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1 + -9t + t2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -9t + -1 + t2 = 0 + -1

Reorder the terms:
1 + -1 + -9t + t2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -9t + t2 = 0 + -1
-9t + t2 = 0 + -1

Combine like terms: 0 + -1 = -1
-9t + t2 = -1

The t term is -9t.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9t + 20.25 + t2 = -1 + 20.25

Reorder the terms:
20.25 + -9t + t2 = -1 + 20.25

Combine like terms: -1 + 20.25 = 19.25
20.25 + -9t + t2 = 19.25

Factor a perfect square on the left side:
(t + -4.5)(t + -4.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(t + -4.5) equal to 4.387482194 and -4.387482194.

Subproblem 1
t + -4.5 = 4.387482194

Simplifying
t + -4.5 = 4.387482194

Reorder the terms:
-4.5 + t = 4.387482194

Solving
-4.5 + t = 4.387482194

Solving for variable 't'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + t = 4.387482194 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + t = 4.387482194 + 4.5
t = 4.387482194 + 4.5

Combine like terms: 4.387482194 + 4.5 = 8.887482194
t = 8.887482194

Simplifying
t = 8.887482194

Subproblem 2
t + -4.5 = -4.387482194

Simplifying
t + -4.5 = -4.387482194

Reorder the terms:
-4.5 + t = -4.387482194

Solving
-4.5 + t = -4.387482194

Solving for variable 't'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + t = -4.387482194 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + t = -4.387482194 + 4.5
t = -4.387482194 + 4.5

Combine like terms: -4.387482194 + 4.5 = 0.112517806
t = 0.112517806

Simplifying
t = 0.112517806

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {8.887482194, 0.112517806}