-6t^2+24t+14=t

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Solution for -6t^2+24t+14=t equation: 


Simplifying
-6t2 + 24t + 14 = t

Reorder the terms:
14 + 24t + -6t2 = t

Solving
14 + 24t + -6t2 = t

Solving for variable 't'.

Reorder the terms:
14 + 24t + -1t + -6t2 = t + -1t

Combine like terms: 24t + -1t = 23t
14 + 23t + -6t2 = t + -1t

Combine like terms: t + -1t = 0
14 + 23t + -6t2 = 0

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

Divide each side by '-6'.
-2.333333333 + -3.833333333t + t2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -3.833333333t + 2.333333333 + t2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -3.833333333t + t2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -3.833333333t + t2 = 0 + 2.333333333
-3.833333333t + t2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-3.833333333t + t2 = 2.333333333

The t term is -3.833333333t.  Take half its coefficient (-1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
-3.833333333t + 3.673611112 + t2 = 2.333333333 + 3.673611112

Reorder the terms:
3.673611112 + -3.833333333t + t2 = 2.333333333 + 3.673611112

Combine like terms: 2.333333333 + 3.673611112 = 6.006944445
3.673611112 + -3.833333333t + t2 = 6.006944445

Factor a perfect square on the left side:
(t + -1.916666667)(t + -1.916666667) = 6.006944445

Calculate the square root of the right side: 2.450906862

Break this problem into two subproblems by setting 
(t + -1.916666667) equal to 2.450906862 and -2.450906862.


Subproblem 1
t + -1.916666667 = 2.450906862

Simplifying
t + -1.916666667 = 2.450906862

Reorder the terms:
-1.916666667 + t = 2.450906862

Solving
-1.916666667 + t = 2.450906862

Solving for variable 't'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + t = 2.450906862 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + t = 2.450906862 + 1.916666667
t = 2.450906862 + 1.916666667

Combine like terms: 2.450906862 + 1.916666667 = 4.367573529
t = 4.367573529

Simplifying
t = 4.367573529


Subproblem 2
t + -1.916666667 = -2.450906862

Simplifying
t + -1.916666667 = -2.450906862

Reorder the terms:
-1.916666667 + t = -2.450906862

Solving
-1.916666667 + t = -2.450906862

Solving for variable 't'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + t = -2.450906862 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + t = -2.450906862 + 1.916666667
t = -2.450906862 + 1.916666667

Combine like terms: -2.450906862 + 1.916666667 = -0.534240195
t = -0.534240195

Simplifying
t = -0.534240195


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {4.367573529, -0.534240195}

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