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Simplifying Z2 + -2Z + 0.75 = 0 Reorder the terms: 0.75 + -2Z + Z2 = 0 Solving 0.75 + -2Z + Z2 = 0 Solving for variable 'Z'. Begin completing the square. Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2Z + -0.75 + Z2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2Z + Z2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2Z + Z2 = 0 + -0.75 -2Z + Z2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2Z + Z2 = -0.75 The Z term is -2Z. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2Z + 1 + Z2 = -0.75 + 1 Reorder the terms: 1 + -2Z + Z2 = -0.75 + 1 Combine like terms: -0.75 + 1 = 0.25 1 + -2Z + Z2 = 0.25 Factor a perfect square on the left side: (Z + -1)(Z + -1) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (Z + -1) equal to 0.5 and -0.5.Subproblem 1
Z + -1 = 0.5 Simplifying Z + -1 = 0.5 Reorder the terms: -1 + Z = 0.5 Solving -1 + Z = 0.5 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + Z = 0.5 + 1 Combine like terms: -1 + 1 = 0 0 + Z = 0.5 + 1 Z = 0.5 + 1 Combine like terms: 0.5 + 1 = 1.5 Z = 1.5 Simplifying Z = 1.5Subproblem 2
Z + -1 = -0.5 Simplifying Z + -1 = -0.5 Reorder the terms: -1 + Z = -0.5 Solving -1 + Z = -0.5 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + Z = -0.5 + 1 Combine like terms: -1 + 1 = 0 0 + Z = -0.5 + 1 Z = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 Z = 0.5 Simplifying Z = 0.5Solution
The solution to the problem is based on the solutions from the subproblems. Z = {1.5, 0.5}
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