2y^2=3y+1

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Solution for 2y^2=3y+1 equation:


Simplifying
2y2 = 3y + 1

Reorder the terms:
2y2 = 1 + 3y

Solving
2y2 = 1 + 3y

Solving for variable 'y'.

Reorder the terms:
-1 + -3y + 2y2 = 1 + 3y + -1 + -3y

Reorder the terms:
-1 + -3y + 2y2 = 1 + -1 + 3y + -3y

Combine like terms: 1 + -1 = 0
-1 + -3y + 2y2 = 0 + 3y + -3y
-1 + -3y + 2y2 = 3y + -3y

Combine like terms: 3y + -3y = 0
-1 + -3y + 2y2 = 0

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

Divide each side by '2'.
-0.5 + -1.5y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1.5y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1.5y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1.5y + y2 = 0 + 0.5
-1.5y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1.5y + y2 = 0.5

The y term is -1.5y.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5y + 0.5625 + y2 = 0.5 + 0.5625

Reorder the terms:
0.5625 + -1.5y + y2 = 0.5 + 0.5625

Combine like terms: 0.5 + 0.5625 = 1.0625
0.5625 + -1.5y + y2 = 1.0625

Factor a perfect square on the left side:
(y + -0.75)(y + -0.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(y + -0.75) equal to 1.030776406 and -1.030776406.

Subproblem 1

y + -0.75 = 1.030776406 Simplifying y + -0.75 = 1.030776406 Reorder the terms: -0.75 + y = 1.030776406 Solving -0.75 + y = 1.030776406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 1.030776406 + 0.75 y = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 y = 1.780776406 Simplifying y = 1.780776406

Subproblem 2

y + -0.75 = -1.030776406 Simplifying y + -0.75 = -1.030776406 Reorder the terms: -0.75 + y = -1.030776406 Solving -0.75 + y = -1.030776406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -1.030776406 + 0.75 y = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 y = -0.280776406 Simplifying y = -0.280776406

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.780776406, -0.280776406}

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