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Simplifying 18y2 + 25y + 5 = 0 Reorder the terms: 5 + 25y + 18y2 = 0 Solving 5 + 25y + 18y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.2777777778 + 1.388888889y + y2 = 0 Move the constant term to the right: Add '-0.2777777778' to each side of the equation. 0.2777777778 + 1.388888889y + -0.2777777778 + y2 = 0 + -0.2777777778 Reorder the terms: 0.2777777778 + -0.2777777778 + 1.388888889y + y2 = 0 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + 1.388888889y + y2 = 0 + -0.2777777778 1.388888889y + y2 = 0 + -0.2777777778 Combine like terms: 0 + -0.2777777778 = -0.2777777778 1.388888889y + y2 = -0.2777777778 The y term is 1.388888889y. Take half its coefficient (0.6944444445). Square it (0.4822530865) and add it to both sides. Add '0.4822530865' to each side of the equation. 1.388888889y + 0.4822530865 + y2 = -0.2777777778 + 0.4822530865 Reorder the terms: 0.4822530865 + 1.388888889y + y2 = -0.2777777778 + 0.4822530865 Combine like terms: -0.2777777778 + 0.4822530865 = 0.2044753087 0.4822530865 + 1.388888889y + y2 = 0.2044753087 Factor a perfect square on the left side: (y + 0.6944444445)(y + 0.6944444445) = 0.2044753087 Calculate the square root of the right side: 0.452189461 Break this problem into two subproblems by setting (y + 0.6944444445) equal to 0.452189461 and -0.452189461.Subproblem 1
y + 0.6944444445 = 0.452189461 Simplifying y + 0.6944444445 = 0.452189461 Reorder the terms: 0.6944444445 + y = 0.452189461 Solving 0.6944444445 + y = 0.452189461 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6944444445' to each side of the equation. 0.6944444445 + -0.6944444445 + y = 0.452189461 + -0.6944444445 Combine like terms: 0.6944444445 + -0.6944444445 = 0.0000000000 0.0000000000 + y = 0.452189461 + -0.6944444445 y = 0.452189461 + -0.6944444445 Combine like terms: 0.452189461 + -0.6944444445 = -0.2422549835 y = -0.2422549835 Simplifying y = -0.2422549835Subproblem 2
y + 0.6944444445 = -0.452189461 Simplifying y + 0.6944444445 = -0.452189461 Reorder the terms: 0.6944444445 + y = -0.452189461 Solving 0.6944444445 + y = -0.452189461 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6944444445' to each side of the equation. 0.6944444445 + -0.6944444445 + y = -0.452189461 + -0.6944444445 Combine like terms: 0.6944444445 + -0.6944444445 = 0.0000000000 0.0000000000 + y = -0.452189461 + -0.6944444445 y = -0.452189461 + -0.6944444445 Combine like terms: -0.452189461 + -0.6944444445 = -1.1466339055 y = -1.1466339055 Simplifying y = -1.1466339055Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.2422549835, -1.1466339055}
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