0=2t^2-6t+3

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


Simplifying
0 = 2t2 + -6t + 3

Reorder the terms:
0 = 3 + -6t + 2t2

Solving
0 = 3 + -6t + 2t2

Solving for variable 't'.

Combine like terms: 0 + -3 = -3
-3 + 6t + -2t2 = 3 + -6t + 2t2 + -3 + 6t + -2t2

Reorder the terms:
-3 + 6t + -2t2 = 3 + -3 + -6t + 6t + 2t2 + -2t2

Combine like terms: 3 + -3 = 0
-3 + 6t + -2t2 = 0 + -6t + 6t + 2t2 + -2t2
-3 + 6t + -2t2 = -6t + 6t + 2t2 + -2t2

Combine like terms: -6t + 6t = 0
-3 + 6t + -2t2 = 0 + 2t2 + -2t2
-3 + 6t + -2t2 = 2t2 + -2t2

Combine like terms: 2t2 + -2t2 = 0
-3 + 6t + -2t2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
1.5 + -3t + t2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3t + -1.5 + t2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3t + t2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3t + t2 = 0 + -1.5
-3t + t2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3t + t2 = -1.5

The t term is -3t.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3t + 2.25 + t2 = -1.5 + 2.25

Reorder the terms:
2.25 + -3t + t2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + -3t + t2 = 0.75

Factor a perfect square on the left side:
(t + -1.5)(t + -1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(t + -1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

t + -1.5 = 0.866025404 Simplifying t + -1.5 = 0.866025404 Reorder the terms: -1.5 + t = 0.866025404 Solving -1.5 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 0.866025404 + 1.5 t = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 t = 2.366025404 Simplifying t = 2.366025404

Subproblem 2

t + -1.5 = -0.866025404 Simplifying t + -1.5 = -0.866025404 Reorder the terms: -1.5 + t = -0.866025404 Solving -1.5 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -0.866025404 + 1.5 t = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 t = 0.633974596 Simplifying t = 0.633974596

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.366025404, 0.633974596}

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