6t^2+4686t-5000=0

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Solution for 6t^2+4686t-5000=0 equation:


Simplifying
6t2 + 4686t + -5000 = 0

Reorder the terms:
-5000 + 4686t + 6t2 = 0

Solving
-5000 + 4686t + 6t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-2500 + 2343t + 3t2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-2500 + 2343t + 3t2)' equal to zero and attempt to solve:

Simplifying
-2500 + 2343t + 3t2 = 0

Solving
-2500 + 2343t + 3t2 = 0

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

Divide each side by '3'.
-833.3333333 + 781t + t2 = 0

Move the constant term to the right:

Add '833.3333333' to each side of the equation.
-833.3333333 + 781t + 833.3333333 + t2 = 0 + 833.3333333

Reorder the terms:
-833.3333333 + 833.3333333 + 781t + t2 = 0 + 833.3333333

Combine like terms: -833.3333333 + 833.3333333 = 0.0000000
0.0000000 + 781t + t2 = 0 + 833.3333333
781t + t2 = 0 + 833.3333333

Combine like terms: 0 + 833.3333333 = 833.3333333
781t + t2 = 833.3333333

The t term is 781t.  Take half its coefficient (390.5).
Square it (152490.25) and add it to both sides.

Add '152490.25' to each side of the equation.
781t + 152490.25 + t2 = 833.3333333 + 152490.25

Reorder the terms:
152490.25 + 781t + t2 = 833.3333333 + 152490.25

Combine like terms: 833.3333333 + 152490.25 = 153323.5833333
152490.25 + 781t + t2 = 153323.5833333

Factor a perfect square on the left side:
(t + 390.5)(t + 390.5) = 153323.5833333

Calculate the square root of the right side: 391.565554324

Break this problem into two subproblems by setting 
(t + 390.5) equal to 391.565554324 and -391.565554324.


Subproblem 1
t + 390.5 = 391.565554324

Simplifying
t + 390.5 = 391.565554324

Reorder the terms:
390.5 + t = 391.565554324

Solving
390.5 + t = 391.565554324

Solving for variable 't'.

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

Add '-390.5' to each side of the equation.
390.5 + -390.5 + t = 391.565554324 + -390.5

Combine like terms: 390.5 + -390.5 = 0.0
0.0 + t = 391.565554324 + -390.5
t = 391.565554324 + -390.5

Combine like terms: 391.565554324 + -390.5 = 1.065554324
t = 1.065554324

Simplifying
t = 1.065554324


Subproblem 2
t + 390.5 = -391.565554324

Simplifying
t + 390.5 = -391.565554324

Reorder the terms:
390.5 + t = -391.565554324

Solving
390.5 + t = -391.565554324

Solving for variable 't'.

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

Add '-390.5' to each side of the equation.
390.5 + -390.5 + t = -391.565554324 + -390.5

Combine like terms: 390.5 + -390.5 = 0.0
0.0 + t = -391.565554324 + -390.5
t = -391.565554324 + -390.5

Combine like terms: -391.565554324 + -390.5 = -782.065554324
t = -782.065554324

Simplifying
t = -782.065554324


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.065554324, -782.065554324}
Solution
t = {1.065554324, -782.065554324}

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