-16t^2+176t-178=0

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Solution for -16t^2+176t-178=0 equation: 


Simplifying
-16t2 + 176t + -178 = 0

Reorder the terms:
-178 + 176t + -16t2 = 0

Solving
-178 + 176t + -16t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-89 + 88t + -8t2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-89 + 88t + -8t2)' equal to zero and attempt to solve:

Simplifying
-89 + 88t + -8t2 = 0

Solving
-89 + 88t + -8t2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
11.125 + -11t + t2 = 0

Move the constant term to the right:

Add '-11.125' to each side of the equation.
11.125 + -11t + -11.125 + t2 = 0 + -11.125

Reorder the terms:
11.125 + -11.125 + -11t + t2 = 0 + -11.125

Combine like terms: 11.125 + -11.125 = 0.000
0.000 + -11t + t2 = 0 + -11.125
-11t + t2 = 0 + -11.125

Combine like terms: 0 + -11.125 = -11.125
-11t + t2 = -11.125

The t term is -11t.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11t + 30.25 + t2 = -11.125 + 30.25

Reorder the terms:
30.25 + -11t + t2 = -11.125 + 30.25

Combine like terms: -11.125 + 30.25 = 19.125
30.25 + -11t + t2 = 19.125

Factor a perfect square on the left side:
(t + -5.5)(t + -5.5) = 19.125

Calculate the square root of the right side: 4.373213921

Break this problem into two subproblems by setting 
(t + -5.5) equal to 4.373213921 and -4.373213921.


Subproblem 1
t + -5.5 = 4.373213921

Simplifying
t + -5.5 = 4.373213921

Reorder the terms:
-5.5 + t = 4.373213921

Solving
-5.5 + t = 4.373213921

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '5.5' to each side of the equation.
-5.5 + 5.5 + t = 4.373213921 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + t = 4.373213921 + 5.5
t = 4.373213921 + 5.5

Combine like terms: 4.373213921 + 5.5 = 9.873213921
t = 9.873213921

Simplifying
t = 9.873213921


Subproblem 2
t + -5.5 = -4.373213921

Simplifying
t + -5.5 = -4.373213921

Reorder the terms:
-5.5 + t = -4.373213921

Solving
-5.5 + t = -4.373213921

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '5.5' to each side of the equation.
-5.5 + 5.5 + t = -4.373213921 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + t = -4.373213921 + 5.5
t = -4.373213921 + 5.5

Combine like terms: -4.373213921 + 5.5 = 1.126786079
t = 1.126786079

Simplifying
t = 1.126786079


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {9.873213921, 1.126786079}
Solution
t = {9.873213921, 1.126786079}

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