23=7+36t-16t^2

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Solution for 23=7+36t-16t^2 equation: 


Simplifying
23 = 7 + 36t + -16t2

Solving
23 = 7 + 36t + -16t2

Solving for variable 't'.

Combine like terms: 23 + -7 = 16
16 + -36t + 16t2 = 7 + 36t + -16t2 + -7 + -36t + 16t2

Reorder the terms:
16 + -36t + 16t2 = 7 + -7 + 36t + -36t + -16t2 + 16t2

Combine like terms: 7 + -7 = 0
16 + -36t + 16t2 = 0 + 36t + -36t + -16t2 + 16t2
16 + -36t + 16t2 = 36t + -36t + -16t2 + 16t2

Combine like terms: 36t + -36t = 0
16 + -36t + 16t2 = 0 + -16t2 + 16t2
16 + -36t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
16 + -36t + 16t2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(4 + -9t + 4t2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(4 + -9t + 4t2)' equal to zero and attempt to solve:

Simplifying
4 + -9t + 4t2 = 0

Solving
4 + -9t + 4t2 = 0

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

Divide each side by '4'.
1 + -2.25t + t2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.25t + -1 + t2 = 0 + -1

Reorder the terms:
1 + -1 + -2.25t + t2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.25t + t2 = 0 + -1
-2.25t + t2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.25t + t2 = -1

The t term is -2.25t.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25t + 1.265625 + t2 = -1 + 1.265625

Reorder the terms:
1.265625 + -2.25t + t2 = -1 + 1.265625

Combine like terms: -1 + 1.265625 = 0.265625
1.265625 + -2.25t + t2 = 0.265625

Factor a perfect square on the left side:
(t + -1.125)(t + -1.125) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(t + -1.125) equal to 0.515388203 and -0.515388203.


Subproblem 1
t + -1.125 = 0.515388203

Simplifying
t + -1.125 = 0.515388203

Reorder the terms:
-1.125 + t = 0.515388203

Solving
-1.125 + t = 0.515388203

Solving for variable 't'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + t = 0.515388203 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + t = 0.515388203 + 1.125
t = 0.515388203 + 1.125

Combine like terms: 0.515388203 + 1.125 = 1.640388203
t = 1.640388203

Simplifying
t = 1.640388203


Subproblem 2
t + -1.125 = -0.515388203

Simplifying
t + -1.125 = -0.515388203

Reorder the terms:
-1.125 + t = -0.515388203

Solving
-1.125 + t = -0.515388203

Solving for variable 't'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + t = -0.515388203 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + t = -0.515388203 + 1.125
t = -0.515388203 + 1.125

Combine like terms: -0.515388203 + 1.125 = 0.609611797
t = 0.609611797

Simplifying
t = 0.609611797


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.640388203, 0.609611797}
Solution
t = {1.640388203, 0.609611797}

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