2t^2-39t+105=0

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Solution for 2t^2-39t+105=0 equation:


Simplifying
2t2 + -39t + 105 = 0

Reorder the terms:
105 + -39t + 2t2 = 0

Solving
105 + -39t + 2t2 = 0

Solving for variable 't'.

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

Divide each side by '2'.
52.5 + -19.5t + t2 = 0

Move the constant term to the right:

Add '-52.5' to each side of the equation.
52.5 + -19.5t + -52.5 + t2 = 0 + -52.5

Reorder the terms:
52.5 + -52.5 + -19.5t + t2 = 0 + -52.5

Combine like terms: 52.5 + -52.5 = 0.0
0.0 + -19.5t + t2 = 0 + -52.5
-19.5t + t2 = 0 + -52.5

Combine like terms: 0 + -52.5 = -52.5
-19.5t + t2 = -52.5

The t term is -19.5t.  Take half its coefficient (-9.75).
Square it (95.0625) and add it to both sides.

Add '95.0625' to each side of the equation.
-19.5t + 95.0625 + t2 = -52.5 + 95.0625

Reorder the terms:
95.0625 + -19.5t + t2 = -52.5 + 95.0625

Combine like terms: -52.5 + 95.0625 = 42.5625
95.0625 + -19.5t + t2 = 42.5625

Factor a perfect square on the left side:
(t + -9.75)(t + -9.75) = 42.5625

Calculate the square root of the right side: 6.523994175

Break this problem into two subproblems by setting 
(t + -9.75) equal to 6.523994175 and -6.523994175.

Subproblem 1

t + -9.75 = 6.523994175 Simplifying t + -9.75 = 6.523994175 Reorder the terms: -9.75 + t = 6.523994175 Solving -9.75 + t = 6.523994175 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9.75' to each side of the equation. -9.75 + 9.75 + t = 6.523994175 + 9.75 Combine like terms: -9.75 + 9.75 = 0.00 0.00 + t = 6.523994175 + 9.75 t = 6.523994175 + 9.75 Combine like terms: 6.523994175 + 9.75 = 16.273994175 t = 16.273994175 Simplifying t = 16.273994175

Subproblem 2

t + -9.75 = -6.523994175 Simplifying t + -9.75 = -6.523994175 Reorder the terms: -9.75 + t = -6.523994175 Solving -9.75 + t = -6.523994175 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9.75' to each side of the equation. -9.75 + 9.75 + t = -6.523994175 + 9.75 Combine like terms: -9.75 + 9.75 = 0.00 0.00 + t = -6.523994175 + 9.75 t = -6.523994175 + 9.75 Combine like terms: -6.523994175 + 9.75 = 3.226005825 t = 3.226005825 Simplifying t = 3.226005825

Solution

The solution to the problem is based on the solutions from the subproblems. t = {16.273994175, 3.226005825}

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