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Simplifying (6y2 + -6y + -5) + -1(2y2 + 3) + -1(5y + 5) = 0 Reorder the terms: (-5 + -6y + 6y2) + -1(2y2 + 3) + -1(5y + 5) = 0 Remove parenthesis around (-5 + -6y + 6y2) -5 + -6y + 6y2 + -1(2y2 + 3) + -1(5y + 5) = 0 Reorder the terms: -5 + -6y + 6y2 + -1(3 + 2y2) + -1(5y + 5) = 0 -5 + -6y + 6y2 + (3 * -1 + 2y2 * -1) + -1(5y + 5) = 0 -5 + -6y + 6y2 + (-3 + -2y2) + -1(5y + 5) = 0 Reorder the terms: -5 + -6y + 6y2 + -3 + -2y2 + -1(5 + 5y) = 0 -5 + -6y + 6y2 + -3 + -2y2 + (5 * -1 + 5y * -1) = 0 -5 + -6y + 6y2 + -3 + -2y2 + (-5 + -5y) = 0 Reorder the terms: -5 + -3 + -5 + -6y + -5y + 6y2 + -2y2 = 0 Combine like terms: -5 + -3 = -8 -8 + -5 + -6y + -5y + 6y2 + -2y2 = 0 Combine like terms: -8 + -5 = -13 -13 + -6y + -5y + 6y2 + -2y2 = 0 Combine like terms: -6y + -5y = -11y -13 + -11y + 6y2 + -2y2 = 0 Combine like terms: 6y2 + -2y2 = 4y2 -13 + -11y + 4y2 = 0 Solving -13 + -11y + 4y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3.25 + -2.75y + y2 = 0 Move the constant term to the right: Add '3.25' to each side of the equation. -3.25 + -2.75y + 3.25 + y2 = 0 + 3.25 Reorder the terms: -3.25 + 3.25 + -2.75y + y2 = 0 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + -2.75y + y2 = 0 + 3.25 -2.75y + y2 = 0 + 3.25 Combine like terms: 0 + 3.25 = 3.25 -2.75y + y2 = 3.25 The y term is -2.75y. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75y + 1.890625 + y2 = 3.25 + 1.890625 Reorder the terms: 1.890625 + -2.75y + y2 = 3.25 + 1.890625 Combine like terms: 3.25 + 1.890625 = 5.140625 1.890625 + -2.75y + y2 = 5.140625 Factor a perfect square on the left side: (y + -1.375)(y + -1.375) = 5.140625 Calculate the square root of the right side: 2.267294643 Break this problem into two subproblems by setting (y + -1.375) equal to 2.267294643 and -2.267294643.Subproblem 1
y + -1.375 = 2.267294643 Simplifying y + -1.375 = 2.267294643 Reorder the terms: -1.375 + y = 2.267294643 Solving -1.375 + y = 2.267294643 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + y = 2.267294643 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + y = 2.267294643 + 1.375 y = 2.267294643 + 1.375 Combine like terms: 2.267294643 + 1.375 = 3.642294643 y = 3.642294643 Simplifying y = 3.642294643Subproblem 2
y + -1.375 = -2.267294643 Simplifying y + -1.375 = -2.267294643 Reorder the terms: -1.375 + y = -2.267294643 Solving -1.375 + y = -2.267294643 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + y = -2.267294643 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + y = -2.267294643 + 1.375 y = -2.267294643 + 1.375 Combine like terms: -2.267294643 + 1.375 = -0.892294643 y = -0.892294643 Simplifying y = -0.892294643Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.642294643, -0.892294643}
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