y^2-4=2y^2-15y+34

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Solution for y^2-4=2y^2-15y+34 equation:


Simplifying
y2 + -4 = 2y2 + -15y + 34

Reorder the terms:
-4 + y2 = 2y2 + -15y + 34

Reorder the terms:
-4 + y2 = 34 + -15y + 2y2

Solving
-4 + y2 = 34 + -15y + 2y2

Solving for variable 'y'.

Reorder the terms:
-4 + -34 + 15y + y2 + -2y2 = 34 + -15y + 2y2 + -34 + 15y + -2y2

Combine like terms: -4 + -34 = -38
-38 + 15y + y2 + -2y2 = 34 + -15y + 2y2 + -34 + 15y + -2y2

Combine like terms: y2 + -2y2 = -1y2
-38 + 15y + -1y2 = 34 + -15y + 2y2 + -34 + 15y + -2y2

Reorder the terms:
-38 + 15y + -1y2 = 34 + -34 + -15y + 15y + 2y2 + -2y2

Combine like terms: 34 + -34 = 0
-38 + 15y + -1y2 = 0 + -15y + 15y + 2y2 + -2y2
-38 + 15y + -1y2 = -15y + 15y + 2y2 + -2y2

Combine like terms: -15y + 15y = 0
-38 + 15y + -1y2 = 0 + 2y2 + -2y2
-38 + 15y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
-38 + 15y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
38 + -15y + y2 = 0

Move the constant term to the right:

Add '-38' to each side of the equation.
38 + -15y + -38 + y2 = 0 + -38

Reorder the terms:
38 + -38 + -15y + y2 = 0 + -38

Combine like terms: 38 + -38 = 0
0 + -15y + y2 = 0 + -38
-15y + y2 = 0 + -38

Combine like terms: 0 + -38 = -38
-15y + y2 = -38

The y term is -15y.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15y + 56.25 + y2 = -38 + 56.25

Reorder the terms:
56.25 + -15y + y2 = -38 + 56.25

Combine like terms: -38 + 56.25 = 18.25
56.25 + -15y + y2 = 18.25

Factor a perfect square on the left side:
(y + -7.5)(y + -7.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(y + -7.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

y + -7.5 = 4.272001873 Simplifying y + -7.5 = 4.272001873 Reorder the terms: -7.5 + y = 4.272001873 Solving -7.5 + y = 4.272001873 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + y = 4.272001873 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + y = 4.272001873 + 7.5 y = 4.272001873 + 7.5 Combine like terms: 4.272001873 + 7.5 = 11.772001873 y = 11.772001873 Simplifying y = 11.772001873

Subproblem 2

y + -7.5 = -4.272001873 Simplifying y + -7.5 = -4.272001873 Reorder the terms: -7.5 + y = -4.272001873 Solving -7.5 + y = -4.272001873 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + y = -4.272001873 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + y = -4.272001873 + 7.5 y = -4.272001873 + 7.5 Combine like terms: -4.272001873 + 7.5 = 3.227998127 y = 3.227998127 Simplifying y = 3.227998127

Solution

The solution to the problem is based on the solutions from the subproblems. y = {11.772001873, 3.227998127}

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