0=6t^2-22t+3

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Solution for 0=6t^2-22t+3 equation:


Simplifying
0 = 6t2 + -22t + 3

Reorder the terms:
0 = 3 + -22t + 6t2

Solving
0 = 3 + -22t + 6t2

Solving for variable 't'.

Combine like terms: 0 + -3 = -3
-3 + 22t + -6t2 = 3 + -22t + 6t2 + -3 + 22t + -6t2

Reorder the terms:
-3 + 22t + -6t2 = 3 + -3 + -22t + 22t + 6t2 + -6t2

Combine like terms: 3 + -3 = 0
-3 + 22t + -6t2 = 0 + -22t + 22t + 6t2 + -6t2
-3 + 22t + -6t2 = -22t + 22t + 6t2 + -6t2

Combine like terms: -22t + 22t = 0
-3 + 22t + -6t2 = 0 + 6t2 + -6t2
-3 + 22t + -6t2 = 6t2 + -6t2

Combine like terms: 6t2 + -6t2 = 0
-3 + 22t + -6t2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
0.5 + -3.666666667t + t2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3.666666667t + -0.5 + t2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3.666666667t + t2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3.666666667t + t2 = 0 + -0.5
-3.666666667t + t2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3.666666667t + t2 = -0.5

The t term is -3.666666667t.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667t + 3.361111114 + t2 = -0.5 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667t + t2 = -0.5 + 3.361111114

Combine like terms: -0.5 + 3.361111114 = 2.861111114
3.361111114 + -3.666666667t + t2 = 2.861111114

Factor a perfect square on the left side:
(t + -1.833333334)(t + -1.833333334) = 2.861111114

Calculate the square root of the right side: 1.691481928

Break this problem into two subproblems by setting 
(t + -1.833333334) equal to 1.691481928 and -1.691481928.

Subproblem 1

t + -1.833333334 = 1.691481928 Simplifying t + -1.833333334 = 1.691481928 Reorder the terms: -1.833333334 + t = 1.691481928 Solving -1.833333334 + t = 1.691481928 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + t = 1.691481928 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = 1.691481928 + 1.833333334 t = 1.691481928 + 1.833333334 Combine like terms: 1.691481928 + 1.833333334 = 3.524815262 t = 3.524815262 Simplifying t = 3.524815262

Subproblem 2

t + -1.833333334 = -1.691481928 Simplifying t + -1.833333334 = -1.691481928 Reorder the terms: -1.833333334 + t = -1.691481928 Solving -1.833333334 + t = -1.691481928 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + t = -1.691481928 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = -1.691481928 + 1.833333334 t = -1.691481928 + 1.833333334 Combine like terms: -1.691481928 + 1.833333334 = 0.141851406 t = 0.141851406 Simplifying t = 0.141851406

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.524815262, 0.141851406}

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