0=105t-16t^2-34

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Solution for 0=105t-16t^2-34 equation:


Simplifying
0 = 105t + -16t2 + -34

Reorder the terms:
0 = -34 + 105t + -16t2

Solving
0 = -34 + 105t + -16t2

Solving for variable 't'.

Combine like terms: 0 + 34 = 34
34 + -105t + 16t2 = -34 + 105t + -16t2 + 34 + -105t + 16t2

Reorder the terms:
34 + -105t + 16t2 = -34 + 34 + 105t + -105t + -16t2 + 16t2

Combine like terms: -34 + 34 = 0
34 + -105t + 16t2 = 0 + 105t + -105t + -16t2 + 16t2
34 + -105t + 16t2 = 105t + -105t + -16t2 + 16t2

Combine like terms: 105t + -105t = 0
34 + -105t + 16t2 = 0 + -16t2 + 16t2
34 + -105t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
34 + -105t + 16t2 = 0

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

Divide each side by '16'.
2.125 + -6.5625t + t2 = 0

Move the constant term to the right:

Add '-2.125' to each side of the equation.
2.125 + -6.5625t + -2.125 + t2 = 0 + -2.125

Reorder the terms:
2.125 + -2.125 + -6.5625t + t2 = 0 + -2.125

Combine like terms: 2.125 + -2.125 = 0.000
0.000 + -6.5625t + t2 = 0 + -2.125
-6.5625t + t2 = 0 + -2.125

Combine like terms: 0 + -2.125 = -2.125
-6.5625t + t2 = -2.125

The t term is -6.5625t.  Take half its coefficient (-3.28125).
Square it (10.76660156) and add it to both sides.

Add '10.76660156' to each side of the equation.
-6.5625t + 10.76660156 + t2 = -2.125 + 10.76660156

Reorder the terms:
10.76660156 + -6.5625t + t2 = -2.125 + 10.76660156

Combine like terms: -2.125 + 10.76660156 = 8.64160156
10.76660156 + -6.5625t + t2 = 8.64160156

Factor a perfect square on the left side:
(t + -3.28125)(t + -3.28125) = 8.64160156

Calculate the square root of the right side: 2.93966011

Break this problem into two subproblems by setting 
(t + -3.28125) equal to 2.93966011 and -2.93966011.

Subproblem 1

t + -3.28125 = 2.93966011 Simplifying t + -3.28125 = 2.93966011 Reorder the terms: -3.28125 + t = 2.93966011 Solving -3.28125 + t = 2.93966011 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.28125' to each side of the equation. -3.28125 + 3.28125 + t = 2.93966011 + 3.28125 Combine like terms: -3.28125 + 3.28125 = 0.00000 0.00000 + t = 2.93966011 + 3.28125 t = 2.93966011 + 3.28125 Combine like terms: 2.93966011 + 3.28125 = 6.22091011 t = 6.22091011 Simplifying t = 6.22091011

Subproblem 2

t + -3.28125 = -2.93966011 Simplifying t + -3.28125 = -2.93966011 Reorder the terms: -3.28125 + t = -2.93966011 Solving -3.28125 + t = -2.93966011 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.28125' to each side of the equation. -3.28125 + 3.28125 + t = -2.93966011 + 3.28125 Combine like terms: -3.28125 + 3.28125 = 0.00000 0.00000 + t = -2.93966011 + 3.28125 t = -2.93966011 + 3.28125 Combine like terms: -2.93966011 + 3.28125 = 0.34158989 t = 0.34158989 Simplifying t = 0.34158989

Solution

The solution to the problem is based on the solutions from the subproblems. t = {6.22091011, 0.34158989}

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