12t(7-3t)=-8t-30

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Solution for 12t(7-3t)=-8t-30 equation: 


Simplifying
12t(7 + -3t) = -8t + -30
(7 * 12t + -3t * 12t) = -8t + -30
(84t + -36t2) = -8t + -30

Reorder the terms:
84t + -36t2 = -30 + -8t

Solving
84t + -36t2 = -30 + -8t

Solving for variable 't'.

Reorder the terms:
30 + 84t + 8t + -36t2 = -30 + -8t + 30 + 8t

Combine like terms: 84t + 8t = 92t
30 + 92t + -36t2 = -30 + -8t + 30 + 8t

Reorder the terms:
30 + 92t + -36t2 = -30 + 30 + -8t + 8t

Combine like terms: -30 + 30 = 0
30 + 92t + -36t2 = 0 + -8t + 8t
30 + 92t + -36t2 = -8t + 8t

Combine like terms: -8t + 8t = 0
30 + 92t + -36t2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(15 + 46t + -18t2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(15 + 46t + -18t2)' equal to zero and attempt to solve:

Simplifying
15 + 46t + -18t2 = 0

Solving
15 + 46t + -18t2 = 0

Begin completing the square.  Divide all terms by
-18 the coefficient of the squared term: 

Divide each side by '-18'.
-0.8333333333 + -2.555555556t + t2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + -2.555555556t + 0.8333333333 + t2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + -2.555555556t + t2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + -2.555555556t + t2 = 0 + 0.8333333333
-2.555555556t + t2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
-2.555555556t + t2 = 0.8333333333

The t term is -2.555555556t.  Take half its coefficient (-1.277777778).
Square it (1.632716050) and add it to both sides.

Add '1.632716050' to each side of the equation.
-2.555555556t + 1.632716050 + t2 = 0.8333333333 + 1.632716050

Reorder the terms:
1.632716050 + -2.555555556t + t2 = 0.8333333333 + 1.632716050

Combine like terms: 0.8333333333 + 1.632716050 = 2.4660493833
1.632716050 + -2.555555556t + t2 = 2.4660493833

Factor a perfect square on the left side:
(t + -1.277777778)(t + -1.277777778) = 2.4660493833

Calculate the square root of the right side: 1.570366003

Break this problem into two subproblems by setting 
(t + -1.277777778) equal to 1.570366003 and -1.570366003.


Subproblem 1
t + -1.277777778 = 1.570366003

Simplifying
t + -1.277777778 = 1.570366003

Reorder the terms:
-1.277777778 + t = 1.570366003

Solving
-1.277777778 + t = 1.570366003

Solving for variable 't'.

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

Add '1.277777778' to each side of the equation.
-1.277777778 + 1.277777778 + t = 1.570366003 + 1.277777778

Combine like terms: -1.277777778 + 1.277777778 = 0.000000000
0.000000000 + t = 1.570366003 + 1.277777778
t = 1.570366003 + 1.277777778

Combine like terms: 1.570366003 + 1.277777778 = 2.848143781
t = 2.848143781

Simplifying
t = 2.848143781


Subproblem 2
t + -1.277777778 = -1.570366003

Simplifying
t + -1.277777778 = -1.570366003

Reorder the terms:
-1.277777778 + t = -1.570366003

Solving
-1.277777778 + t = -1.570366003

Solving for variable 't'.

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

Add '1.277777778' to each side of the equation.
-1.277777778 + 1.277777778 + t = -1.570366003 + 1.277777778

Combine like terms: -1.277777778 + 1.277777778 = 0.000000000
0.000000000 + t = -1.570366003 + 1.277777778
t = -1.570366003 + 1.277777778

Combine like terms: -1.570366003 + 1.277777778 = -0.292588225
t = -0.292588225

Simplifying
t = -0.292588225


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.848143781, -0.292588225}
Solution
t = {2.848143781, -0.292588225}

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