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Simplifying 0.00000000000025 = 0.00000000000000864 + 0.000000000144y + 0.000006y2 Solving 0.00000000000025 = 0.00000000000000864 + 0.000000000144y + 0.000006y2 Solving for variable 'y'. Combine like terms: 0.00000000000025 + -0.00000000000000864 = 0.00000000000024136 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.00000000000000864 + 0.000000000144y + 0.000006y2 + -0.00000000000000864 + -0.000000000144y + -0.000006y2 Reorder the terms: 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.00000000000000864 + -0.00000000000000864 + 0.000000000144y + -0.000000000144y + 0.000006y2 + -0.000006y2 Combine like terms: 0.00000000000000864 + -0.00000000000000864 = 0.00000000000000000 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.00000000000000000 + 0.000000000144y + -0.000000000144y + 0.000006y2 + -0.000006y2 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.000000000144y + -0.000000000144y + 0.000006y2 + -0.000006y2 Combine like terms: 0.000000000144y + -0.000000000144y = 0.000000000000 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.000000000000 + 0.000006y2 + -0.000006y2 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.000006y2 + -0.000006y2 Combine like terms: 0.000006y2 + -0.000006y2 = 0.000000 0.00000000000024136 + -0.000000000144y + -0.000006y2 = 0.000000 Begin completing the square. Divide all terms by -0.000006 the coefficient of the squared term: Divide each side by '-0.000006'. -0.00000004022666667 + 0.000024y + y2 = 0 Move the constant term to the right: Add '0.00000004022666667' to each side of the equation. -0.00000004022666667 + 0.000024y + 0.00000004022666667 + y2 = 0 + 0.00000004022666667 Reorder the terms: -0.00000004022666667 + 0.00000004022666667 + 0.000024y + y2 = 0 + 0.00000004022666667 Combine like terms: -0.00000004022666667 + 0.00000004022666667 = 0.00000000000000000 0.00000000000000000 + 0.000024y + y2 = 0 + 0.00000004022666667 0.000024y + y2 = 0 + 0.00000004022666667 Combine like terms: 0 + 0.00000004022666667 = 0.00000004022666667 0.000024y + y2 = 0.00000004022666667 The y term is 0.000024y. Take half its coefficient (0.000012). Square it (0.000000000144) and add it to both sides. Add '0.000000000144' to each side of the equation. 0.000024y + 0.000000000144 + y2 = 0.00000004022666667 + 0.000000000144 Reorder the terms: 0.000000000144 + 0.000024y + y2 = 0.00000004022666667 + 0.000000000144 Combine like terms: 0.00000004022666667 + 0.000000000144 = 0.00000004037066667 0.000000000144 + 0.000024y + y2 = 0.00000004037066667 Factor a perfect square on the left side: (y + 0.000012)(y + 0.000012) = 0.00000004037066667 Calculate the square root of the right side: 0.000200925 Break this problem into two subproblems by setting (y + 0.000012) equal to 0.000200925 and -0.000200925.Subproblem 1
y + 0.000012 = 0.000200925 Simplifying y + 0.000012 = 0.000200925 Reorder the terms: 0.000012 + y = 0.000200925 Solving 0.000012 + y = 0.000200925 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.000012' to each side of the equation. 0.000012 + -0.000012 + y = 0.000200925 + -0.000012 Combine like terms: 0.000012 + -0.000012 = 0.000000 0.000000 + y = 0.000200925 + -0.000012 y = 0.000200925 + -0.000012 Combine like terms: 0.000200925 + -0.000012 = 0.000188925 y = 0.000188925 Simplifying y = 0.000188925Subproblem 2
y + 0.000012 = -0.000200925 Simplifying y + 0.000012 = -0.000200925 Reorder the terms: 0.000012 + y = -0.000200925 Solving 0.000012 + y = -0.000200925 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.000012' to each side of the equation. 0.000012 + -0.000012 + y = -0.000200925 + -0.000012 Combine like terms: 0.000012 + -0.000012 = 0.000000 0.000000 + y = -0.000200925 + -0.000012 y = -0.000200925 + -0.000012 Combine like terms: -0.000200925 + -0.000012 = -0.000212925 y = -0.000212925 Simplifying y = -0.000212925Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.000188925, -0.000212925}
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