9t(t+2)+-3t(6t+1)=1

Solution for 9t(t+2)+-3t(6t+1)=1 equation:

Simplifying
9t(t + 2) + -3t(6t + 1) = 1

Reorder the terms:
9t(2 + t) + -3t(6t + 1) = 1
(2 * 9t + t * 9t) + -3t(6t + 1) = 1
(18t + 9t2) + -3t(6t + 1) = 1

Reorder the terms:
18t + 9t2 + -3t(1 + 6t) = 1
18t + 9t2 + (1 * -3t + 6t * -3t) = 1
18t + 9t2 + (-3t + -18t2) = 1

Reorder the terms:
18t + -3t + 9t2 + -18t2 = 1

Combine like terms: 18t + -3t = 15t
15t + 9t2 + -18t2 = 1

Combine like terms: 9t2 + -18t2 = -9t2
15t + -9t2 = 1

Solving
15t + -9t2 = 1

Solving for variable 't'.

Reorder the terms:
-1 + 15t + -9t2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 15t + -9t2 = 0

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
0.1111111111 + -1.666666667t + t2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -1.666666667t + -0.1111111111 + t2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + -1.666666667t + t2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + -1.666666667t + t2 = 0 + -0.1111111111
-1.666666667t + t2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
-1.666666667t + t2 = -0.1111111111

The t term is -1.666666667t.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667t + 0.6944444447 + t2 = -0.1111111111 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667t + t2 = -0.1111111111 + 0.6944444447

Combine like terms: -0.1111111111 + 0.6944444447 = 0.5833333336
0.6944444447 + -1.666666667t + t2 = 0.5833333336

Factor a perfect square on the left side:
(t + -0.8333333335)(t + -0.8333333335) = 0.5833333336

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(t + -0.8333333335) equal to 0.763762616 and -0.763762616.

Subproblem 1
t + -0.8333333335 = 0.763762616

Simplifying
t + -0.8333333335 = 0.763762616

Reorder the terms:
-0.8333333335 + t = 0.763762616

Solving
-0.8333333335 + t = 0.763762616

Solving for variable 't'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + t = 0.763762616 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + t = 0.763762616 + 0.8333333335
t = 0.763762616 + 0.8333333335

Combine like terms: 0.763762616 + 0.8333333335 = 1.5970959495
t = 1.5970959495

Simplifying
t = 1.5970959495

Subproblem 2
t + -0.8333333335 = -0.763762616

Simplifying
t + -0.8333333335 = -0.763762616

Reorder the terms:
-0.8333333335 + t = -0.763762616

Solving
-0.8333333335 + t = -0.763762616

Solving for variable 't'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + t = -0.763762616 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + t = -0.763762616 + 0.8333333335
t = -0.763762616 + 0.8333333335

Combine like terms: -0.763762616 + 0.8333333335 = 0.0695707175
t = 0.0695707175

Simplifying
t = 0.0695707175

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.5970959495, 0.0695707175}