2y^2-5y+3y^2-100-120y+36y^2=8

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Solution for 2y^2-5y+3y^2-100-120y+36y^2=8 equation:


Simplifying
2y2 + -5y + 3y2 + -100 + -120y + 36y2 = 8

Reorder the terms:
-100 + -5y + -120y + 2y2 + 3y2 + 36y2 = 8

Combine like terms: -5y + -120y = -125y
-100 + -125y + 2y2 + 3y2 + 36y2 = 8

Combine like terms: 2y2 + 3y2 = 5y2
-100 + -125y + 5y2 + 36y2 = 8

Combine like terms: 5y2 + 36y2 = 41y2
-100 + -125y + 41y2 = 8

Solving
-100 + -125y + 41y2 = 8

Solving for variable 'y'.

Reorder the terms:
-100 + -8 + -125y + 41y2 = 8 + -8

Combine like terms: -100 + -8 = -108
-108 + -125y + 41y2 = 8 + -8

Combine like terms: 8 + -8 = 0
-108 + -125y + 41y2 = 0

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

Divide each side by '41'.
-2.634146341 + -3.048780488y + y2 = 0

Move the constant term to the right:

Add '2.634146341' to each side of the equation.
-2.634146341 + -3.048780488y + 2.634146341 + y2 = 0 + 2.634146341

Reorder the terms:
-2.634146341 + 2.634146341 + -3.048780488y + y2 = 0 + 2.634146341

Combine like terms: -2.634146341 + 2.634146341 = 0.000000000
0.000000000 + -3.048780488y + y2 = 0 + 2.634146341
-3.048780488y + y2 = 0 + 2.634146341

Combine like terms: 0 + 2.634146341 = 2.634146341
-3.048780488y + y2 = 2.634146341

The y term is -3.048780488y.  Take half its coefficient (-1.524390244).
Square it (2.323765616) and add it to both sides.

Add '2.323765616' to each side of the equation.
-3.048780488y + 2.323765616 + y2 = 2.634146341 + 2.323765616

Reorder the terms:
2.323765616 + -3.048780488y + y2 = 2.634146341 + 2.323765616

Combine like terms: 2.634146341 + 2.323765616 = 4.957911957
2.323765616 + -3.048780488y + y2 = 4.957911957

Factor a perfect square on the left side:
(y + -1.524390244)(y + -1.524390244) = 4.957911957

Calculate the square root of the right side: 2.226636916

Break this problem into two subproblems by setting 
(y + -1.524390244) equal to 2.226636916 and -2.226636916.

Subproblem 1

y + -1.524390244 = 2.226636916 Simplifying y + -1.524390244 = 2.226636916 Reorder the terms: -1.524390244 + y = 2.226636916 Solving -1.524390244 + y = 2.226636916 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.524390244' to each side of the equation. -1.524390244 + 1.524390244 + y = 2.226636916 + 1.524390244 Combine like terms: -1.524390244 + 1.524390244 = 0.000000000 0.000000000 + y = 2.226636916 + 1.524390244 y = 2.226636916 + 1.524390244 Combine like terms: 2.226636916 + 1.524390244 = 3.75102716 y = 3.75102716 Simplifying y = 3.75102716

Subproblem 2

y + -1.524390244 = -2.226636916 Simplifying y + -1.524390244 = -2.226636916 Reorder the terms: -1.524390244 + y = -2.226636916 Solving -1.524390244 + y = -2.226636916 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.524390244' to each side of the equation. -1.524390244 + 1.524390244 + y = -2.226636916 + 1.524390244 Combine like terms: -1.524390244 + 1.524390244 = 0.000000000 0.000000000 + y = -2.226636916 + 1.524390244 y = -2.226636916 + 1.524390244 Combine like terms: -2.226636916 + 1.524390244 = -0.702246672 y = -0.702246672 Simplifying y = -0.702246672

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.75102716, -0.702246672}

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