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