0=-16t^2+52t+3

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Solution for 0=-16t^2+52t+3 equation:


Simplifying
0 = -16t2 + 52t + 3

Reorder the terms:
0 = 3 + 52t + -16t2

Solving
0 = 3 + 52t + -16t2

Solving for variable 't'.

Combine like terms: 0 + -3 = -3
-3 + -52t + 16t2 = 3 + 52t + -16t2 + -3 + -52t + 16t2

Reorder the terms:
-3 + -52t + 16t2 = 3 + -3 + 52t + -52t + -16t2 + 16t2

Combine like terms: 3 + -3 = 0
-3 + -52t + 16t2 = 0 + 52t + -52t + -16t2 + 16t2
-3 + -52t + 16t2 = 52t + -52t + -16t2 + 16t2

Combine like terms: 52t + -52t = 0
-3 + -52t + 16t2 = 0 + -16t2 + 16t2
-3 + -52t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-3 + -52t + 16t2 = 0

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

Divide each side by '16'.
-0.1875 + -3.25t + t2 = 0

Move the constant term to the right:

Add '0.1875' to each side of the equation.
-0.1875 + -3.25t + 0.1875 + t2 = 0 + 0.1875

Reorder the terms:
-0.1875 + 0.1875 + -3.25t + t2 = 0 + 0.1875

Combine like terms: -0.1875 + 0.1875 = 0.0000
0.0000 + -3.25t + t2 = 0 + 0.1875
-3.25t + t2 = 0 + 0.1875

Combine like terms: 0 + 0.1875 = 0.1875
-3.25t + t2 = 0.1875

The t term is -3.25t.  Take half its coefficient (-1.625).
Square it (2.640625) and add it to both sides.

Add '2.640625' to each side of the equation.
-3.25t + 2.640625 + t2 = 0.1875 + 2.640625

Reorder the terms:
2.640625 + -3.25t + t2 = 0.1875 + 2.640625

Combine like terms: 0.1875 + 2.640625 = 2.828125
2.640625 + -3.25t + t2 = 2.828125

Factor a perfect square on the left side:
(t + -1.625)(t + -1.625) = 2.828125

Calculate the square root of the right side: 1.681703006

Break this problem into two subproblems by setting 
(t + -1.625) equal to 1.681703006 and -1.681703006.

Subproblem 1

t + -1.625 = 1.681703006 Simplifying t + -1.625 = 1.681703006 Reorder the terms: -1.625 + t = 1.681703006 Solving -1.625 + t = 1.681703006 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + t = 1.681703006 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + t = 1.681703006 + 1.625 t = 1.681703006 + 1.625 Combine like terms: 1.681703006 + 1.625 = 3.306703006 t = 3.306703006 Simplifying t = 3.306703006

Subproblem 2

t + -1.625 = -1.681703006 Simplifying t + -1.625 = -1.681703006 Reorder the terms: -1.625 + t = -1.681703006 Solving -1.625 + t = -1.681703006 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + t = -1.681703006 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + t = -1.681703006 + 1.625 t = -1.681703006 + 1.625 Combine like terms: -1.681703006 + 1.625 = -0.056703006 t = -0.056703006 Simplifying t = -0.056703006

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.306703006, -0.056703006}

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