-6t^2+32t+42=0

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Solution for -6t^2+32t+42=0 equation: 


Simplifying
-6t2 + 32t + 42 = 0

Reorder the terms:
42 + 32t + -6t2 = 0

Solving
42 + 32t + -6t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(21 + 16t + -3t2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(21 + 16t + -3t2)' equal to zero and attempt to solve:

Simplifying
21 + 16t + -3t2 = 0

Solving
21 + 16t + -3t2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-7 + -5.333333333t + t2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -5.333333333t + 7 + t2 = 0 + 7

Reorder the terms:
-7 + 7 + -5.333333333t + t2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -5.333333333t + t2 = 0 + 7
-5.333333333t + t2 = 0 + 7

Combine like terms: 0 + 7 = 7
-5.333333333t + t2 = 7

The t term is -5.333333333t.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333t + 7.111111113 + t2 = 7 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333t + t2 = 7 + 7.111111113

Combine like terms: 7 + 7.111111113 = 14.111111113
7.111111113 + -5.333333333t + t2 = 14.111111113

Factor a perfect square on the left side:
(t + -2.666666667)(t + -2.666666667) = 14.111111113

Calculate the square root of the right side: 3.75647589

Break this problem into two subproblems by setting 
(t + -2.666666667) equal to 3.75647589 and -3.75647589.


Subproblem 1
t + -2.666666667 = 3.75647589

Simplifying
t + -2.666666667 = 3.75647589

Reorder the terms:
-2.666666667 + t = 3.75647589

Solving
-2.666666667 + t = 3.75647589

Solving for variable 't'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + t = 3.75647589 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + t = 3.75647589 + 2.666666667
t = 3.75647589 + 2.666666667

Combine like terms: 3.75647589 + 2.666666667 = 6.423142557
t = 6.423142557

Simplifying
t = 6.423142557


Subproblem 2
t + -2.666666667 = -3.75647589

Simplifying
t + -2.666666667 = -3.75647589

Reorder the terms:
-2.666666667 + t = -3.75647589

Solving
-2.666666667 + t = -3.75647589

Solving for variable 't'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + t = -3.75647589 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + t = -3.75647589 + 2.666666667
t = -3.75647589 + 2.666666667

Combine like terms: -3.75647589 + 2.666666667 = -1.089809223
t = -1.089809223

Simplifying
t = -1.089809223


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {6.423142557, -1.089809223}
Solution
t = {6.423142557, -1.089809223}

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