53=-16t^2+75t+30

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Solution for 53=-16t^2+75t+30 equation:


Simplifying
53 = -16t2 + 75t + 30

Reorder the terms:
53 = 30 + 75t + -16t2

Solving
53 = 30 + 75t + -16t2

Solving for variable 't'.

Combine like terms: 53 + -30 = 23
23 + -75t + 16t2 = 30 + 75t + -16t2 + -30 + -75t + 16t2

Reorder the terms:
23 + -75t + 16t2 = 30 + -30 + 75t + -75t + -16t2 + 16t2

Combine like terms: 30 + -30 = 0
23 + -75t + 16t2 = 0 + 75t + -75t + -16t2 + 16t2
23 + -75t + 16t2 = 75t + -75t + -16t2 + 16t2

Combine like terms: 75t + -75t = 0
23 + -75t + 16t2 = 0 + -16t2 + 16t2
23 + -75t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
23 + -75t + 16t2 = 0

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

Divide each side by '16'.
1.4375 + -4.6875t + t2 = 0

Move the constant term to the right:

Add '-1.4375' to each side of the equation.
1.4375 + -4.6875t + -1.4375 + t2 = 0 + -1.4375

Reorder the terms:
1.4375 + -1.4375 + -4.6875t + t2 = 0 + -1.4375

Combine like terms: 1.4375 + -1.4375 = 0.0000
0.0000 + -4.6875t + t2 = 0 + -1.4375
-4.6875t + t2 = 0 + -1.4375

Combine like terms: 0 + -1.4375 = -1.4375
-4.6875t + t2 = -1.4375

The t term is -4.6875t.  Take half its coefficient (-2.34375).
Square it (5.493164063) and add it to both sides.

Add '5.493164063' to each side of the equation.
-4.6875t + 5.493164063 + t2 = -1.4375 + 5.493164063

Reorder the terms:
5.493164063 + -4.6875t + t2 = -1.4375 + 5.493164063

Combine like terms: -1.4375 + 5.493164063 = 4.055664063
5.493164063 + -4.6875t + t2 = 4.055664063

Factor a perfect square on the left side:
(t + -2.34375)(t + -2.34375) = 4.055664063

Calculate the square root of the right side: 2.013867936

Break this problem into two subproblems by setting 
(t + -2.34375) equal to 2.013867936 and -2.013867936.

Subproblem 1

t + -2.34375 = 2.013867936 Simplifying t + -2.34375 = 2.013867936 Reorder the terms: -2.34375 + t = 2.013867936 Solving -2.34375 + t = 2.013867936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.34375' to each side of the equation. -2.34375 + 2.34375 + t = 2.013867936 + 2.34375 Combine like terms: -2.34375 + 2.34375 = 0.00000 0.00000 + t = 2.013867936 + 2.34375 t = 2.013867936 + 2.34375 Combine like terms: 2.013867936 + 2.34375 = 4.357617936 t = 4.357617936 Simplifying t = 4.357617936

Subproblem 2

t + -2.34375 = -2.013867936 Simplifying t + -2.34375 = -2.013867936 Reorder the terms: -2.34375 + t = -2.013867936 Solving -2.34375 + t = -2.013867936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.34375' to each side of the equation. -2.34375 + 2.34375 + t = -2.013867936 + 2.34375 Combine like terms: -2.34375 + 2.34375 = 0.00000 0.00000 + t = -2.013867936 + 2.34375 t = -2.013867936 + 2.34375 Combine like terms: -2.013867936 + 2.34375 = 0.329882064 t = 0.329882064 Simplifying t = 0.329882064

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.357617936, 0.329882064}

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