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Simplifying 2Y2 + 15Y + 10 = 0 Reorder the terms: 10 + 15Y + 2Y2 = 0 Solving 10 + 15Y + 2Y2 = 0 Solving for variable 'Y'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 5 + 7.5Y + Y2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + 7.5Y + -5 + Y2 = 0 + -5 Reorder the terms: 5 + -5 + 7.5Y + Y2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 7.5Y + Y2 = 0 + -5 7.5Y + Y2 = 0 + -5 Combine like terms: 0 + -5 = -5 7.5Y + Y2 = -5 The Y term is 7.5Y. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5Y + 14.0625 + Y2 = -5 + 14.0625 Reorder the terms: 14.0625 + 7.5Y + Y2 = -5 + 14.0625 Combine like terms: -5 + 14.0625 = 9.0625 14.0625 + 7.5Y + Y2 = 9.0625 Factor a perfect square on the left side: (Y + 3.75)(Y + 3.75) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (Y + 3.75) equal to 3.010398645 and -3.010398645.Subproblem 1
Y + 3.75 = 3.010398645 Simplifying Y + 3.75 = 3.010398645 Reorder the terms: 3.75 + Y = 3.010398645 Solving 3.75 + Y = 3.010398645 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + Y = 3.010398645 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + Y = 3.010398645 + -3.75 Y = 3.010398645 + -3.75 Combine like terms: 3.010398645 + -3.75 = -0.739601355 Y = -0.739601355 Simplifying Y = -0.739601355Subproblem 2
Y + 3.75 = -3.010398645 Simplifying Y + 3.75 = -3.010398645 Reorder the terms: 3.75 + Y = -3.010398645 Solving 3.75 + Y = -3.010398645 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + Y = -3.010398645 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + Y = -3.010398645 + -3.75 Y = -3.010398645 + -3.75 Combine like terms: -3.010398645 + -3.75 = -6.760398645 Y = -6.760398645 Simplifying Y = -6.760398645Solution
The solution to the problem is based on the solutions from the subproblems. Y = {-0.739601355, -6.760398645}
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