24t^2+3t-6=0

Solution for 24t^2+3t-6=0 equation:

Simplifying
24t2 + 3t + -6 = 0

Reorder the terms:
-6 + 3t + 24t2 = 0

Solving
-6 + 3t + 24t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-2 + t + 8t2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-2 + t + 8t2)' equal to zero and attempt to solve:

Simplifying
-2 + t + 8t2 = 0

Solving
-2 + t + 8t2 = 0

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

Divide each side by '8'.
-0.25 + 0.125t + t2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.125t + 0.25 + t2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.125t + t2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.125t + t2 = 0 + 0.25
0.125t + t2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.125t + t2 = 0.25

The t term is t.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.125t + 0.25 + t2 = 0.25 + 0.25

Reorder the terms:
0.25 + 0.125t + t2 = 0.25 + 0.25

Combine like terms: 0.25 + 0.25 = 0.5
0.25 + 0.125t + t2 = 0.5

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(t + 0.5) equal to 0.707106781 and -0.707106781.

Subproblem 1
t + 0.5 = 0.707106781

Simplifying
t + 0.5 = 0.707106781

Reorder the terms:
0.5 + t = 0.707106781

Solving
0.5 + t = 0.707106781

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 0.707106781 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 0.707106781 + -0.5
t = 0.707106781 + -0.5

Combine like terms: 0.707106781 + -0.5 = 0.207106781
t = 0.207106781

Simplifying
t = 0.207106781

Subproblem 2
t + 0.5 = -0.707106781

Simplifying
t + 0.5 = -0.707106781

Reorder the terms:
0.5 + t = -0.707106781

Solving
0.5 + t = -0.707106781

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -0.707106781 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -0.707106781 + -0.5
t = -0.707106781 + -0.5

Combine like terms: -0.707106781 + -0.5 = -1.207106781
t = -1.207106781

Simplifying
t = -1.207106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.207106781, -1.207106781}
Solution
t = {0.207106781, -1.207106781}