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Simplifying (y + 1)(y + -1) + 2y = 0 Reorder the terms: (1 + y)(y + -1) + 2y = 0 Reorder the terms: (1 + y)(-1 + y) + 2y = 0 Multiply (1 + y) * (-1 + y) (1(-1 + y) + y(-1 + y)) + 2y = 0 ((-1 * 1 + y * 1) + y(-1 + y)) + 2y = 0 ((-1 + 1y) + y(-1 + y)) + 2y = 0 (-1 + 1y + (-1 * y + y * y)) + 2y = 0 (-1 + 1y + (-1y + y2)) + 2y = 0 Combine like terms: 1y + -1y = 0 (-1 + 0 + y2) + 2y = 0 (-1 + y2) + 2y = 0 Reorder the terms: -1 + 2y + y2 = 0 Solving -1 + 2y + y2 = 0 Solving for variable 'y'. Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 2y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + 2y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2y + y2 = 0 + 1 2y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 2y + y2 = 1 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 1 + 1 Reorder the terms: 1 + 2y + y2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + 2y + y2 = 2 Factor a perfect square on the left side: (y + 1)(y + 1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (y + 1) equal to 1.414213562 and -1.414213562.Subproblem 1
y + 1 = 1.414213562 Simplifying y + 1 = 1.414213562 Reorder the terms: 1 + y = 1.414213562 Solving 1 + y = 1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + y = 1.414213562 + -1 y = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 y = 0.414213562 Simplifying y = 0.414213562Subproblem 2
y + 1 = -1.414213562 Simplifying y + 1 = -1.414213562 Reorder the terms: 1 + y = -1.414213562 Solving 1 + y = -1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + y = -1.414213562 + -1 y = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 y = -2.414213562 Simplifying y = -2.414213562Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.414213562, -2.414213562}
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