432=-16t^2+16t+512

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Solution for 432=-16t^2+16t+512 equation: 


Simplifying
432 = -16t2 + 16t + 512

Reorder the terms:
432 = 512 + 16t + -16t2

Solving
432 = 512 + 16t + -16t2

Solving for variable 't'.

Combine like terms: 432 + -512 = -80
-80 + -16t + 16t2 = 512 + 16t + -16t2 + -512 + -16t + 16t2

Reorder the terms:
-80 + -16t + 16t2 = 512 + -512 + 16t + -16t + -16t2 + 16t2

Combine like terms: 512 + -512 = 0
-80 + -16t + 16t2 = 0 + 16t + -16t + -16t2 + 16t2
-80 + -16t + 16t2 = 16t + -16t + -16t2 + 16t2

Combine like terms: 16t + -16t = 0
-80 + -16t + 16t2 = 0 + -16t2 + 16t2
-80 + -16t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-80 + -16t + 16t2 = 0

Factor out the Greatest Common Factor (GCF), '16'.
16(-5 + -1t + t2) = 0

Ignore the factor 16.

Subproblem 1
Set the factor '(-5 + -1t + t2)' equal to zero and attempt to solve:

Simplifying
-5 + -1t + t2 = 0

Solving
-5 + -1t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1t + 5 + t2 = 0 + 5

Reorder the terms:
-5 + 5 + -1t + t2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1t + t2 = 0 + 5
-1t + t2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1t + t2 = 5

The t term is -1t.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1t + 0.25 + t2 = 5 + 0.25

Reorder the terms:
0.25 + -1t + t2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1t + t2 = 5.25

Factor a perfect square on the left side:
(t + -0.5)(t + -0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(t + -0.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
t + -0.5 = 2.291287847

Simplifying
t + -0.5 = 2.291287847

Reorder the terms:
-0.5 + t = 2.291287847

Solving
-0.5 + t = 2.291287847

Solving for variable 't'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + t = 2.291287847 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + t = 2.291287847 + 0.5
t = 2.291287847 + 0.5

Combine like terms: 2.291287847 + 0.5 = 2.791287847
t = 2.791287847

Simplifying
t = 2.791287847


Subproblem 2
t + -0.5 = -2.291287847

Simplifying
t + -0.5 = -2.291287847

Reorder the terms:
-0.5 + t = -2.291287847

Solving
-0.5 + t = -2.291287847

Solving for variable 't'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + t = -2.291287847 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + t = -2.291287847 + 0.5
t = -2.291287847 + 0.5

Combine like terms: -2.291287847 + 0.5 = -1.791287847
t = -1.791287847

Simplifying
t = -1.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.791287847, -1.791287847}
Solution
t = {2.791287847, -1.791287847}

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