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Simplifying 10400 = 2.6y(10 + 4000y + -1 + -10) Reorder the terms: 10400 = 2.6y(10 + -1 + -10 + 4000y) Combine like terms: 10 + -1 = 9 10400 = 2.6y(9 + -10 + 4000y) Combine like terms: 9 + -10 = -1 10400 = 2.6y(-1 + 4000y) 10400 = (-1 * 2.6y + 4000y * 2.6y) 10400 = (-2.6y + 10400y2) Solving 10400 = -2.6y + 10400y2 Solving for variable 'y'. Reorder the terms: 10400 + 2.6y + -10400y2 = -2.6y + 2.6y + 10400y2 + -10400y2 Combine like terms: -2.6y + 2.6y = 0.0 10400 + 2.6y + -10400y2 = 0.0 + 10400y2 + -10400y2 10400 + 2.6y + -10400y2 = 10400y2 + -10400y2 Combine like terms: 10400y2 + -10400y2 = 0 10400 + 2.6y + -10400y2 = 0 Begin completing the square. Divide all terms by -10400 the coefficient of the squared term: Divide each side by '-10400'. -1 + -0.00025y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.00025y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + -0.00025y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.00025y + y2 = 0 + 1 -0.00025y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.00025y + y2 = 1 The y term is -0.00025y. Take half its coefficient (-0.000125). Square it (0.000000015625) and add it to both sides. Add '0.000000015625' to each side of the equation. -0.00025y + 0.000000015625 + y2 = 1 + 0.000000015625 Reorder the terms: 0.000000015625 + -0.00025y + y2 = 1 + 0.000000015625 Combine like terms: 1 + 0.000000015625 = 1.000000015625 0.000000015625 + -0.00025y + y2 = 1.000000015625 Factor a perfect square on the left side: (y + -0.000125)(y + -0.000125) = 1.000000015625 Calculate the square root of the right side: 1.000000008 Break this problem into two subproblems by setting (y + -0.000125) equal to 1.000000008 and -1.000000008.Subproblem 1
y + -0.000125 = 1.000000008 Simplifying y + -0.000125 = 1.000000008 Reorder the terms: -0.000125 + y = 1.000000008 Solving -0.000125 + y = 1.000000008 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.000125' to each side of the equation. -0.000125 + 0.000125 + y = 1.000000008 + 0.000125 Combine like terms: -0.000125 + 0.000125 = 0.000000 0.000000 + y = 1.000000008 + 0.000125 y = 1.000000008 + 0.000125 Combine like terms: 1.000000008 + 0.000125 = 1.000125008 y = 1.000125008 Simplifying y = 1.000125008Subproblem 2
y + -0.000125 = -1.000000008 Simplifying y + -0.000125 = -1.000000008 Reorder the terms: -0.000125 + y = -1.000000008 Solving -0.000125 + y = -1.000000008 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.000125' to each side of the equation. -0.000125 + 0.000125 + y = -1.000000008 + 0.000125 Combine like terms: -0.000125 + 0.000125 = 0.000000 0.000000 + y = -1.000000008 + 0.000125 y = -1.000000008 + 0.000125 Combine like terms: -1.000000008 + 0.000125 = -0.999875008 y = -0.999875008 Simplifying y = -0.999875008Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.000125008, -0.999875008}
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