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

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


Simplifying
t = -6t2 + 24t + 14

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

Solving
t = 14 + 24t + -6t2

Solving for variable 't'.

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

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

Reorder the terms:
-14 + -23t + 6t2 = 14 + -14 + 24t + -24t + -6t2 + 6t2

Combine like terms: 14 + -14 = 0
-14 + -23t + 6t2 = 0 + 24t + -24t + -6t2 + 6t2
-14 + -23t + 6t2 = 24t + -24t + -6t2 + 6t2

Combine like terms: 24t + -24t = 0
-14 + -23t + 6t2 = 0 + -6t2 + 6t2
-14 + -23t + 6t2 = -6t2 + 6t2

Combine like terms: -6t2 + 6t2 = 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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