t^2+3t+1=0

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

Simplifying
t2 + 3t + 1 = 0

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

Solving
1 + 3t + t2 = 0

Solving for variable 't'.

Begin completing the square.

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 = 1.25
2.25 + 3t + t2 = 1.25

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(t + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
t + 1.5 = 1.118033989

Simplifying
t + 1.5 = 1.118033989

Reorder the terms:
1.5 + t = 1.118033989

Solving
1.5 + t = 1.118033989

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 = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = 1.118033989 + -1.5
t = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
t = -0.381966011

Simplifying
t = -0.381966011

Subproblem 2
t + 1.5 = -1.118033989

Simplifying
t + 1.5 = -1.118033989

Reorder the terms:
1.5 + t = -1.118033989

Solving
1.5 + t = -1.118033989

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 = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = -1.118033989 + -1.5
t = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
t = -2.618033989

Simplifying
t = -2.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-0.381966011, -2.618033989}