0=-16t^2+35t+4.5

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Solution for 0=-16t^2+35t+4.5 equation:


Simplifying
0 = -16t2 + 35t + 4.5

Reorder the terms:
0 = 4.5 + 35t + -16t2

Solving
0 = 4.5 + 35t + -16t2

Solving for variable 't'.

Combine like terms: 0 + -4.5 = -4.5
-4.5 + -35t + 16t2 = 4.5 + 35t + -16t2 + -4.5 + -35t + 16t2

Reorder the terms:
-4.5 + -35t + 16t2 = 4.5 + -4.5 + 35t + -35t + -16t2 + 16t2

Combine like terms: 4.5 + -4.5 = 0.0
-4.5 + -35t + 16t2 = 0.0 + 35t + -35t + -16t2 + 16t2
-4.5 + -35t + 16t2 = 35t + -35t + -16t2 + 16t2

Combine like terms: 35t + -35t = 0
-4.5 + -35t + 16t2 = 0 + -16t2 + 16t2
-4.5 + -35t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-4.5 + -35t + 16t2 = 0

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

Divide each side by '16'.
-0.28125 + -2.1875t + t2 = 0

Move the constant term to the right:

Add '0.28125' to each side of the equation.
-0.28125 + -2.1875t + 0.28125 + t2 = 0 + 0.28125

Reorder the terms:
-0.28125 + 0.28125 + -2.1875t + t2 = 0 + 0.28125

Combine like terms: -0.28125 + 0.28125 = 0.00000
0.00000 + -2.1875t + t2 = 0 + 0.28125
-2.1875t + t2 = 0 + 0.28125

Combine like terms: 0 + 0.28125 = 0.28125
-2.1875t + t2 = 0.28125

The t term is -2.1875t.  Take half its coefficient (-1.09375).
Square it (1.196289063) and add it to both sides.

Add '1.196289063' to each side of the equation.
-2.1875t + 1.196289063 + t2 = 0.28125 + 1.196289063

Reorder the terms:
1.196289063 + -2.1875t + t2 = 0.28125 + 1.196289063

Combine like terms: 0.28125 + 1.196289063 = 1.477539063
1.196289063 + -2.1875t + t2 = 1.477539063

Factor a perfect square on the left side:
(t + -1.09375)(t + -1.09375) = 1.477539063

Calculate the square root of the right side: 1.215540646

Break this problem into two subproblems by setting 
(t + -1.09375) equal to 1.215540646 and -1.215540646.

Subproblem 1

t + -1.09375 = 1.215540646 Simplifying t + -1.09375 = 1.215540646 Reorder the terms: -1.09375 + t = 1.215540646 Solving -1.09375 + t = 1.215540646 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + t = 1.215540646 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + t = 1.215540646 + 1.09375 t = 1.215540646 + 1.09375 Combine like terms: 1.215540646 + 1.09375 = 2.309290646 t = 2.309290646 Simplifying t = 2.309290646

Subproblem 2

t + -1.09375 = -1.215540646 Simplifying t + -1.09375 = -1.215540646 Reorder the terms: -1.09375 + t = -1.215540646 Solving -1.09375 + t = -1.215540646 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + t = -1.215540646 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + t = -1.215540646 + 1.09375 t = -1.215540646 + 1.09375 Combine like terms: -1.215540646 + 1.09375 = -0.121790646 t = -0.121790646 Simplifying t = -0.121790646

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.309290646, -0.121790646}

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