3t^2+5t-16=0

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Solution for 3t^2+5t-16=0 equation: 


Simplifying
3t2 + 5t + -16 = 0

Reorder the terms:
-16 + 5t + 3t2 = 0

Solving
-16 + 5t + 3t2 = 0

Solving for variable 't'.

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

Divide each side by '3'.
-5.333333333 + 1.666666667t + t2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + 1.666666667t + 5.333333333 + t2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + 1.666666667t + t2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + 1.666666667t + t2 = 0 + 5.333333333
1.666666667t + t2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
1.666666667t + t2 = 5.333333333

The t term is 1.666666667t.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667t + 0.6944444447 + t2 = 5.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667t + t2 = 5.333333333 + 0.6944444447

Combine like terms: 5.333333333 + 0.6944444447 = 6.0277777777
0.6944444447 + 1.666666667t + t2 = 6.0277777777

Factor a perfect square on the left side:
(t + 0.8333333335)(t + 0.8333333335) = 6.0277777777

Calculate the square root of the right side: 2.45515331

Break this problem into two subproblems by setting 
(t + 0.8333333335) equal to 2.45515331 and -2.45515331.


Subproblem 1
t + 0.8333333335 = 2.45515331

Simplifying
t + 0.8333333335 = 2.45515331

Reorder the terms:
0.8333333335 + t = 2.45515331

Solving
0.8333333335 + t = 2.45515331

Solving for variable 't'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + t = 2.45515331 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + t = 2.45515331 + -0.8333333335
t = 2.45515331 + -0.8333333335

Combine like terms: 2.45515331 + -0.8333333335 = 1.6218199765
t = 1.6218199765

Simplifying
t = 1.6218199765


Subproblem 2
t + 0.8333333335 = -2.45515331

Simplifying
t + 0.8333333335 = -2.45515331

Reorder the terms:
0.8333333335 + t = -2.45515331

Solving
0.8333333335 + t = -2.45515331

Solving for variable 't'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + t = -2.45515331 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + t = -2.45515331 + -0.8333333335
t = -2.45515331 + -0.8333333335

Combine like terms: -2.45515331 + -0.8333333335 = -3.2884866435
t = -3.2884866435

Simplifying
t = -3.2884866435


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.6218199765, -3.2884866435}

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