576=16t^2+32t+560

Solution for 576=16t^2+32t+560 equation:

Simplifying
576 = 16t2 + 32t + 560

Reorder the terms:
576 = 560 + 32t + 16t2

Solving
576 = 560 + 32t + 16t2

Solving for variable 't'.

Combine like terms: 576 + -560 = 16
16 + -32t + -16t2 = 560 + 32t + 16t2 + -560 + -32t + -16t2

Reorder the terms:
16 + -32t + -16t2 = 560 + -560 + 32t + -32t + 16t2 + -16t2

Combine like terms: 560 + -560 = 0
16 + -32t + -16t2 = 0 + 32t + -32t + 16t2 + -16t2
16 + -32t + -16t2 = 32t + -32t + 16t2 + -16t2

Combine like terms: 32t + -32t = 0
16 + -32t + -16t2 = 0 + 16t2 + -16t2
16 + -32t + -16t2 = 16t2 + -16t2

Combine like terms: 16t2 + -16t2 = 0
16 + -32t + -16t2 = 0

Factor out the Greatest Common Factor (GCF), '16'.
16(1 + -2t + -1t2) = 0

Ignore the factor 16.
Subproblem 1
Set the factor '(1 + -2t + -1t2)' equal to zero and attempt to solve:

Simplifying
1 + -2t + -1t2 = 0

Solving
1 + -2t + -1t2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + 2t + t2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2t + 1 + t2 = 0 + 1

Reorder the terms:
-1 + 1 + 2t + t2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2t + t2 = 0 + 1
2t + t2 = 0 + 1

Combine like terms: 0 + 1 = 1
2t + t2 = 1

The t term is 2t.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2t + 1 + t2 = 1 + 1

Reorder the terms:
1 + 2t + t2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + 2t + t2 = 2

Factor a perfect square on the left side:
(t + 1)(t + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(t + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
t + 1 = 1.414213562

Simplifying
t + 1 = 1.414213562

Reorder the terms:
1 + t = 1.414213562

Solving
1 + t = 1.414213562

Solving for variable 't'.

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

Add '-1' to each side of the equation.
1 + -1 + t = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + t = 1.414213562 + -1
t = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
t = 0.414213562

Simplifying
t = 0.414213562

Subproblem 2
t + 1 = -1.414213562

Simplifying
t + 1 = -1.414213562

Reorder the terms:
1 + t = -1.414213562

Solving
1 + t = -1.414213562

Solving for variable 't'.

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

Add '-1' to each side of the equation.
1 + -1 + t = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + t = -1.414213562 + -1
t = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
t = -2.414213562

Simplifying
t = -2.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.414213562, -2.414213562}
Solution
t = {0.414213562, -2.414213562}