12t^2+19t=6

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Solution for 12t^2+19t=6 equation:


Simplifying
12t2 + 19t = 6

Reorder the terms:
19t + 12t2 = 6

Solving
19t + 12t2 = 6

Solving for variable 't'.

Reorder the terms:
-6 + 19t + 12t2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 19t + 12t2 = 0

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

Divide each side by '12'.
-0.5 + 1.583333333t + t2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 1.583333333t + 0.5 + t2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 1.583333333t + t2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 1.583333333t + t2 = 0 + 0.5
1.583333333t + t2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
1.583333333t + t2 = 0.5

The t term is 1.583333333t.  Take half its coefficient (0.7916666665).
Square it (0.6267361108) and add it to both sides.

Add '0.6267361108' to each side of the equation.
1.583333333t + 0.6267361108 + t2 = 0.5 + 0.6267361108

Reorder the terms:
0.6267361108 + 1.583333333t + t2 = 0.5 + 0.6267361108

Combine like terms: 0.5 + 0.6267361108 = 1.1267361108
0.6267361108 + 1.583333333t + t2 = 1.1267361108

Factor a perfect square on the left side:
(t + 0.7916666665)(t + 0.7916666665) = 1.1267361108

Calculate the square root of the right side: 1.061478267

Break this problem into two subproblems by setting 
(t + 0.7916666665) equal to 1.061478267 and -1.061478267.

Subproblem 1

t + 0.7916666665 = 1.061478267 Simplifying t + 0.7916666665 = 1.061478267 Reorder the terms: 0.7916666665 + t = 1.061478267 Solving 0.7916666665 + t = 1.061478267 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.7916666665' to each side of the equation. 0.7916666665 + -0.7916666665 + t = 1.061478267 + -0.7916666665 Combine like terms: 0.7916666665 + -0.7916666665 = 0.0000000000 0.0000000000 + t = 1.061478267 + -0.7916666665 t = 1.061478267 + -0.7916666665 Combine like terms: 1.061478267 + -0.7916666665 = 0.2698116005 t = 0.2698116005 Simplifying t = 0.2698116005

Subproblem 2

t + 0.7916666665 = -1.061478267 Simplifying t + 0.7916666665 = -1.061478267 Reorder the terms: 0.7916666665 + t = -1.061478267 Solving 0.7916666665 + t = -1.061478267 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.7916666665' to each side of the equation. 0.7916666665 + -0.7916666665 + t = -1.061478267 + -0.7916666665 Combine like terms: 0.7916666665 + -0.7916666665 = 0.0000000000 0.0000000000 + t = -1.061478267 + -0.7916666665 t = -1.061478267 + -0.7916666665 Combine like terms: -1.061478267 + -0.7916666665 = -1.8531449335 t = -1.8531449335 Simplifying t = -1.8531449335

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.2698116005, -1.8531449335}

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