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Simplifying 4y + 4 = 14y + 14 + -2y2 Reorder the terms: 4 + 4y = 14y + 14 + -2y2 Reorder the terms: 4 + 4y = 14 + 14y + -2y2 Solving 4 + 4y = 14 + 14y + -2y2 Solving for variable 'y'. Reorder the terms: 4 + -14 + 4y + -14y + 2y2 = 14 + 14y + -2y2 + -14 + -14y + 2y2 Combine like terms: 4 + -14 = -10 -10 + 4y + -14y + 2y2 = 14 + 14y + -2y2 + -14 + -14y + 2y2 Combine like terms: 4y + -14y = -10y -10 + -10y + 2y2 = 14 + 14y + -2y2 + -14 + -14y + 2y2 Reorder the terms: -10 + -10y + 2y2 = 14 + -14 + 14y + -14y + -2y2 + 2y2 Combine like terms: 14 + -14 = 0 -10 + -10y + 2y2 = 0 + 14y + -14y + -2y2 + 2y2 -10 + -10y + 2y2 = 14y + -14y + -2y2 + 2y2 Combine like terms: 14y + -14y = 0 -10 + -10y + 2y2 = 0 + -2y2 + 2y2 -10 + -10y + 2y2 = -2y2 + 2y2 Combine like terms: -2y2 + 2y2 = 0 -10 + -10y + 2y2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -5y + y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -5y + y2)' equal to zero and attempt to solve: Simplifying -5 + -5y + y2 = 0 Solving -5 + -5y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -5y + 5 + y2 = 0 + 5 Reorder the terms: -5 + 5 + -5y + y2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -5y + y2 = 0 + 5 -5y + y2 = 0 + 5 Combine like terms: 0 + 5 = 5 -5y + y2 = 5 The y term is -5y. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5y + 6.25 + y2 = 5 + 6.25 Reorder the terms: 6.25 + -5y + y2 = 5 + 6.25 Combine like terms: 5 + 6.25 = 11.25 6.25 + -5y + y2 = 11.25 Factor a perfect square on the left side: (y + -2.5)(y + -2.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (y + -2.5) equal to 3.354101966 and -3.354101966.Subproblem 1
y + -2.5 = 3.354101966 Simplifying y + -2.5 = 3.354101966 Reorder the terms: -2.5 + y = 3.354101966 Solving -2.5 + y = 3.354101966 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = 3.354101966 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = 3.354101966 + 2.5 y = 3.354101966 + 2.5 Combine like terms: 3.354101966 + 2.5 = 5.854101966 y = 5.854101966 Simplifying y = 5.854101966Subproblem 2
y + -2.5 = -3.354101966 Simplifying y + -2.5 = -3.354101966 Reorder the terms: -2.5 + y = -3.354101966 Solving -2.5 + y = -3.354101966 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = -3.354101966 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = -3.354101966 + 2.5 y = -3.354101966 + 2.5 Combine like terms: -3.354101966 + 2.5 = -0.854101966 y = -0.854101966 Simplifying y = -0.854101966Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.854101966, -0.854101966}Solution
y = {5.854101966, -0.854101966}
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