2y^2-5=-2y

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Solution for 2y^2-5=-2y equation:


Simplifying
2y2 + -5 = -2y

Reorder the terms:
-5 + 2y2 = -2y

Solving
-5 + 2y2 = -2y

Solving for variable 'y'.

Reorder the terms:
-5 + 2y + 2y2 = -2y + 2y

Combine like terms: -2y + 2y = 0
-5 + 2y + 2y2 = 0

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

Divide each side by '2'.
-2.5 + y + y2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + y + 2.5 + y2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + y + y2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + y + y2 = 0 + 2.5
y + y2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
y + y2 = 2.5

The y term is y.  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.
 + 0.25 + y2 = 2.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + y2 = 2.5 + 0.25

Combine like terms: 2.5 + 0.25 = 2.75
1.25 + y2 = 2.75

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(y + 0.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

y + 0.5 = 1.658312395 Simplifying y + 0.5 = 1.658312395 Reorder the terms: 0.5 + y = 1.658312395 Solving 0.5 + y = 1.658312395 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.658312395 + -0.5 y = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 y = 1.158312395 Simplifying y = 1.158312395

Subproblem 2

y + 0.5 = -1.658312395 Simplifying y + 0.5 = -1.658312395 Reorder the terms: 0.5 + y = -1.658312395 Solving 0.5 + y = -1.658312395 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.658312395 + -0.5 y = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 y = -2.158312395 Simplifying y = -2.158312395

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.158312395, -2.158312395}

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