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Simplifying z2 + -1z = z + 1 Reorder the terms: -1z + z2 = z + 1 Reorder the terms: -1z + z2 = 1 + z Solving -1z + z2 = 1 + z Solving for variable 'z'. Reorder the terms: -1 + -1z + -1z + z2 = 1 + z + -1 + -1z Combine like terms: -1z + -1z = -2z -1 + -2z + z2 = 1 + z + -1 + -1z Reorder the terms: -1 + -2z + z2 = 1 + -1 + z + -1z Combine like terms: 1 + -1 = 0 -1 + -2z + z2 = 0 + z + -1z -1 + -2z + z2 = z + -1z Combine like terms: z + -1z = 0 -1 + -2z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -2z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + -2z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2z + z2 = 0 + 1 -2z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2z + z2 = 1 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 = 1 + 1 Reorder the terms: 1 + -2z + z2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + -2z + z2 = 2 Factor a perfect square on the left side: (z + -1)(z + -1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (z + -1) equal to 1.414213562 and -1.414213562.Subproblem 1
z + -1 = 1.414213562 Simplifying z + -1 = 1.414213562 Reorder the terms: -1 + z = 1.414213562 Solving -1 + z = 1.414213562 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 = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + z = 1.414213562 + 1 z = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 z = 2.414213562 Simplifying z = 2.414213562Subproblem 2
z + -1 = -1.414213562 Simplifying z + -1 = -1.414213562 Reorder the terms: -1 + z = -1.414213562 Solving -1 + z = -1.414213562 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 = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + z = -1.414213562 + 1 z = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 z = -0.414213562 Simplifying z = -0.414213562Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.414213562, -0.414213562}
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