8=5+23t-16t^2

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Solution for 8=5+23t-16t^2 equation:


Simplifying
8 = 5 + 23t + -16t2

Solving
8 = 5 + 23t + -16t2

Solving for variable 't'.

Combine like terms: 8 + -5 = 3
3 + -23t + 16t2 = 5 + 23t + -16t2 + -5 + -23t + 16t2

Reorder the terms:
3 + -23t + 16t2 = 5 + -5 + 23t + -23t + -16t2 + 16t2

Combine like terms: 5 + -5 = 0
3 + -23t + 16t2 = 0 + 23t + -23t + -16t2 + 16t2
3 + -23t + 16t2 = 23t + -23t + -16t2 + 16t2

Combine like terms: 23t + -23t = 0
3 + -23t + 16t2 = 0 + -16t2 + 16t2
3 + -23t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
3 + -23t + 16t2 = 0

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

Divide each side by '16'.
0.1875 + -1.4375t + t2 = 0

Move the constant term to the right:

Add '-0.1875' to each side of the equation.
0.1875 + -1.4375t + -0.1875 + t2 = 0 + -0.1875

Reorder the terms:
0.1875 + -0.1875 + -1.4375t + t2 = 0 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + -1.4375t + t2 = 0 + -0.1875
-1.4375t + t2 = 0 + -0.1875

Combine like terms: 0 + -0.1875 = -0.1875
-1.4375t + t2 = -0.1875

The t term is -1.4375t.  Take half its coefficient (-0.71875).
Square it (0.5166015625) and add it to both sides.

Add '0.5166015625' to each side of the equation.
-1.4375t + 0.5166015625 + t2 = -0.1875 + 0.5166015625

Reorder the terms:
0.5166015625 + -1.4375t + t2 = -0.1875 + 0.5166015625

Combine like terms: -0.1875 + 0.5166015625 = 0.3291015625
0.5166015625 + -1.4375t + t2 = 0.3291015625

Factor a perfect square on the left side:
(t + -0.71875)(t + -0.71875) = 0.3291015625

Calculate the square root of the right side: 0.573673742

Break this problem into two subproblems by setting 
(t + -0.71875) equal to 0.573673742 and -0.573673742.

Subproblem 1

t + -0.71875 = 0.573673742 Simplifying t + -0.71875 = 0.573673742 Reorder the terms: -0.71875 + t = 0.573673742 Solving -0.71875 + t = 0.573673742 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + t = 0.573673742 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = 0.573673742 + 0.71875 t = 0.573673742 + 0.71875 Combine like terms: 0.573673742 + 0.71875 = 1.292423742 t = 1.292423742 Simplifying t = 1.292423742

Subproblem 2

t + -0.71875 = -0.573673742 Simplifying t + -0.71875 = -0.573673742 Reorder the terms: -0.71875 + t = -0.573673742 Solving -0.71875 + t = -0.573673742 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + t = -0.573673742 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = -0.573673742 + 0.71875 t = -0.573673742 + 0.71875 Combine like terms: -0.573673742 + 0.71875 = 0.145076258 t = 0.145076258 Simplifying t = 0.145076258

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.292423742, 0.145076258}

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