0=-16t^2+88t-67

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Solution for 0=-16t^2+88t-67 equation:


Simplifying
0 = -16t2 + 88t + -67

Reorder the terms:
0 = -67 + 88t + -16t2

Solving
0 = -67 + 88t + -16t2

Solving for variable 't'.

Combine like terms: 0 + 67 = 67
67 + -88t + 16t2 = -67 + 88t + -16t2 + 67 + -88t + 16t2

Reorder the terms:
67 + -88t + 16t2 = -67 + 67 + 88t + -88t + -16t2 + 16t2

Combine like terms: -67 + 67 = 0
67 + -88t + 16t2 = 0 + 88t + -88t + -16t2 + 16t2
67 + -88t + 16t2 = 88t + -88t + -16t2 + 16t2

Combine like terms: 88t + -88t = 0
67 + -88t + 16t2 = 0 + -16t2 + 16t2
67 + -88t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
67 + -88t + 16t2 = 0

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

Divide each side by '16'.
4.1875 + -5.5t + t2 = 0

Move the constant term to the right:

Add '-4.1875' to each side of the equation.
4.1875 + -5.5t + -4.1875 + t2 = 0 + -4.1875

Reorder the terms:
4.1875 + -4.1875 + -5.5t + t2 = 0 + -4.1875

Combine like terms: 4.1875 + -4.1875 = 0.0000
0.0000 + -5.5t + t2 = 0 + -4.1875
-5.5t + t2 = 0 + -4.1875

Combine like terms: 0 + -4.1875 = -4.1875
-5.5t + t2 = -4.1875

The t term is -5.5t.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5t + 7.5625 + t2 = -4.1875 + 7.5625

Reorder the terms:
7.5625 + -5.5t + t2 = -4.1875 + 7.5625

Combine like terms: -4.1875 + 7.5625 = 3.375
7.5625 + -5.5t + t2 = 3.375

Factor a perfect square on the left side:
(t + -2.75)(t + -2.75) = 3.375

Calculate the square root of the right side: 1.837117307

Break this problem into two subproblems by setting 
(t + -2.75) equal to 1.837117307 and -1.837117307.

Subproblem 1

t + -2.75 = 1.837117307 Simplifying t + -2.75 = 1.837117307 Reorder the terms: -2.75 + t = 1.837117307 Solving -2.75 + t = 1.837117307 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + t = 1.837117307 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + t = 1.837117307 + 2.75 t = 1.837117307 + 2.75 Combine like terms: 1.837117307 + 2.75 = 4.587117307 t = 4.587117307 Simplifying t = 4.587117307

Subproblem 2

t + -2.75 = -1.837117307 Simplifying t + -2.75 = -1.837117307 Reorder the terms: -2.75 + t = -1.837117307 Solving -2.75 + t = -1.837117307 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + t = -1.837117307 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + t = -1.837117307 + 2.75 t = -1.837117307 + 2.75 Combine like terms: -1.837117307 + 2.75 = 0.912882693 t = 0.912882693 Simplifying t = 0.912882693

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.587117307, 0.912882693}

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