12t^2+24t+10=0

Solution for 12t^2+24t+10=0 equation:

Simplifying
12t2 + 24t + 10 = 0

Reorder the terms:
10 + 24t + 12t2 = 0

Solving
10 + 24t + 12t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + 12t + 6t2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(5 + 12t + 6t2)' equal to zero and attempt to solve:

Simplifying
5 + 12t + 6t2 = 0

Solving
5 + 12t + 6t2 = 0

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

Divide each side by '6'.
0.8333333333 + 2t + t2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + 2t + -0.8333333333 + t2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + 2t + t2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + 2t + t2 = 0 + -0.8333333333
2t + t2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
2t + t2 = -0.8333333333

The t term is 2t.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2t + 1 + t2 = -0.8333333333 + 1

Reorder the terms:
1 + 2t + t2 = -0.8333333333 + 1

Combine like terms: -0.8333333333 + 1 = 0.1666666667
1 + 2t + t2 = 0.1666666667

Factor a perfect square on the left side:
(t + 1)(t + 1) = 0.1666666667

Calculate the square root of the right side: 0.408248291

Break this problem into two subproblems by setting 
(t + 1) equal to 0.408248291 and -0.408248291.

Subproblem 1
t + 1 = 0.408248291

Simplifying
t + 1 = 0.408248291

Reorder the terms:
1 + t = 0.408248291

Solving
1 + t = 0.408248291

Solving for variable 't'.

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

Add '-1' to each side of the equation.
1 + -1 + t = 0.408248291 + -1

Combine like terms: 1 + -1 = 0
0 + t = 0.408248291 + -1
t = 0.408248291 + -1

Combine like terms: 0.408248291 + -1 = -0.591751709
t = -0.591751709

Simplifying
t = -0.591751709

Subproblem 2
t + 1 = -0.408248291

Simplifying
t + 1 = -0.408248291

Reorder the terms:
1 + t = -0.408248291

Solving
1 + t = -0.408248291

Solving for variable 't'.

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

Add '-1' to each side of the equation.
1 + -1 + t = -0.408248291 + -1

Combine like terms: 1 + -1 = 0
0 + t = -0.408248291 + -1
t = -0.408248291 + -1

Combine like terms: -0.408248291 + -1 = -1.408248291
t = -1.408248291

Simplifying
t = -1.408248291

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-0.591751709, -1.408248291}
Solution
t = {-0.591751709, -1.408248291}