443=-16t^2-4t+470

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Solution for 443=-16t^2-4t+470 equation:


Simplifying
443 = -16t2 + -4t + 470

Reorder the terms:
443 = 470 + -4t + -16t2

Solving
443 = 470 + -4t + -16t2

Solving for variable 't'.

Combine like terms: 443 + -470 = -27
-27 + 4t + 16t2 = 470 + -4t + -16t2 + -470 + 4t + 16t2

Reorder the terms:
-27 + 4t + 16t2 = 470 + -470 + -4t + 4t + -16t2 + 16t2

Combine like terms: 470 + -470 = 0
-27 + 4t + 16t2 = 0 + -4t + 4t + -16t2 + 16t2
-27 + 4t + 16t2 = -4t + 4t + -16t2 + 16t2

Combine like terms: -4t + 4t = 0
-27 + 4t + 16t2 = 0 + -16t2 + 16t2
-27 + 4t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-27 + 4t + 16t2 = 0

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

Divide each side by '16'.
-1.6875 + 0.25t + t2 = 0

Move the constant term to the right:

Add '1.6875' to each side of the equation.
-1.6875 + 0.25t + 1.6875 + t2 = 0 + 1.6875

Reorder the terms:
-1.6875 + 1.6875 + 0.25t + t2 = 0 + 1.6875

Combine like terms: -1.6875 + 1.6875 = 0.0000
0.0000 + 0.25t + t2 = 0 + 1.6875
0.25t + t2 = 0 + 1.6875

Combine like terms: 0 + 1.6875 = 1.6875
0.25t + t2 = 1.6875

The t term is 0.25t.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25t + 0.015625 + t2 = 1.6875 + 0.015625

Reorder the terms:
0.015625 + 0.25t + t2 = 1.6875 + 0.015625

Combine like terms: 1.6875 + 0.015625 = 1.703125
0.015625 + 0.25t + t2 = 1.703125

Factor a perfect square on the left side:
(t + 0.125)(t + 0.125) = 1.703125

Calculate the square root of the right side: 1.305038314

Break this problem into two subproblems by setting 
(t + 0.125) equal to 1.305038314 and -1.305038314.

Subproblem 1

t + 0.125 = 1.305038314 Simplifying t + 0.125 = 1.305038314 Reorder the terms: 0.125 + t = 1.305038314 Solving 0.125 + t = 1.305038314 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = 1.305038314 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = 1.305038314 + -0.125 t = 1.305038314 + -0.125 Combine like terms: 1.305038314 + -0.125 = 1.180038314 t = 1.180038314 Simplifying t = 1.180038314

Subproblem 2

t + 0.125 = -1.305038314 Simplifying t + 0.125 = -1.305038314 Reorder the terms: 0.125 + t = -1.305038314 Solving 0.125 + t = -1.305038314 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = -1.305038314 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = -1.305038314 + -0.125 t = -1.305038314 + -0.125 Combine like terms: -1.305038314 + -0.125 = -1.430038314 t = -1.430038314 Simplifying t = -1.430038314

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.180038314, -1.430038314}

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