-t^2-3t+1=0

Solution for -t^2-3t+1=0 equation:

Simplifying
-1t2 + -3t + 1 = 0

Reorder the terms:
1 + -3t + -1t2 = 0

Solving
1 + -3t + -1t2 = 0

Solving for variable 't'.

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

Divide each side by '-1'.
-1 + 3t + t2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3t + 1 + t2 = 0 + 1

Reorder the terms:
-1 + 1 + 3t + t2 = 0 + 1

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

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

The t term is 3t.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3t + 2.25 + t2 = 1 + 2.25

Reorder the terms:
2.25 + 3t + t2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + 3t + t2 = 3.25

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(t + 1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
t + 1.5 = 1.802775638

Simplifying
t + 1.5 = 1.802775638

Reorder the terms:
1.5 + t = 1.802775638

Solving
1.5 + t = 1.802775638

Solving for variable 't'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + t = 1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = 1.802775638 + -1.5
t = 1.802775638 + -1.5

Combine like terms: 1.802775638 + -1.5 = 0.302775638
t = 0.302775638

Simplifying
t = 0.302775638

Subproblem 2
t + 1.5 = -1.802775638

Simplifying
t + 1.5 = -1.802775638

Reorder the terms:
1.5 + t = -1.802775638

Solving
1.5 + t = -1.802775638

Solving for variable 't'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + t = -1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = -1.802775638 + -1.5
t = -1.802775638 + -1.5

Combine like terms: -1.802775638 + -1.5 = -3.302775638
t = -3.302775638

Simplifying
t = -3.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.302775638, -3.302775638}