3t^2-16t-3=0

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Solution for 3t^2-16t-3=0 equation:


Simplifying
3t2 + -16t + -3 = 0

Reorder the terms:
-3 + -16t + 3t2 = 0

Solving
-3 + -16t + 3t2 = 0

Solving for variable 't'.

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

Divide each side by '3'.
-1 + -5.333333333t + t2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -5.333333333t + 1 + t2 = 0 + 1

Reorder the terms:
-1 + 1 + -5.333333333t + t2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -5.333333333t + t2 = 0 + 1
-5.333333333t + t2 = 0 + 1

Combine like terms: 0 + 1 = 1
-5.333333333t + t2 = 1

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 = 1 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333t + t2 = 1 + 7.111111113

Combine like terms: 1 + 7.111111113 = 8.111111113
7.111111113 + -5.333333333t + t2 = 8.111111113

Factor a perfect square on the left side:
(t + -2.666666667)(t + -2.666666667) = 8.111111113

Calculate the square root of the right side: 2.848001249

Break this problem into two subproblems by setting 
(t + -2.666666667) equal to 2.848001249 and -2.848001249.

Subproblem 1

t + -2.666666667 = 2.848001249 Simplifying t + -2.666666667 = 2.848001249 Reorder the terms: -2.666666667 + t = 2.848001249 Solving -2.666666667 + t = 2.848001249 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 = 2.848001249 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + t = 2.848001249 + 2.666666667 t = 2.848001249 + 2.666666667 Combine like terms: 2.848001249 + 2.666666667 = 5.514667916 t = 5.514667916 Simplifying t = 5.514667916

Subproblem 2

t + -2.666666667 = -2.848001249 Simplifying t + -2.666666667 = -2.848001249 Reorder the terms: -2.666666667 + t = -2.848001249 Solving -2.666666667 + t = -2.848001249 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 = -2.848001249 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + t = -2.848001249 + 2.666666667 t = -2.848001249 + 2.666666667 Combine like terms: -2.848001249 + 2.666666667 = -0.181334582 t = -0.181334582 Simplifying t = -0.181334582

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.514667916, -0.181334582}

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