2t^2=2t+1

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Solution for 2t^2=2t+1 equation:


Simplifying
2t2 = 2t + 1

Reorder the terms:
2t2 = 1 + 2t

Solving
2t2 = 1 + 2t

Solving for variable 't'.

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

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

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

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

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

Divide each side by '2'.
-0.5 + -1t + t2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1t + 0.5 + t2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1t + t2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1t + t2 = 0 + 0.5
-1t + t2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1t + t2 = 0.5

The t term is -1t.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1t + 0.25 + t2 = 0.5 + 0.25

Reorder the terms:
0.25 + -1t + t2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
0.25 + -1t + t2 = 0.75

Factor a perfect square on the left side:
(t + -0.5)(t + -0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(t + -0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

t + -0.5 = 0.866025404 Simplifying t + -0.5 = 0.866025404 Reorder the terms: -0.5 + t = 0.866025404 Solving -0.5 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 0.866025404 + 0.5 t = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 t = 1.366025404 Simplifying t = 1.366025404

Subproblem 2

t + -0.5 = -0.866025404 Simplifying t + -0.5 = -0.866025404 Reorder the terms: -0.5 + t = -0.866025404 Solving -0.5 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -0.866025404 + 0.5 t = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 t = -0.366025404 Simplifying t = -0.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.366025404, -0.366025404}

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