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Simplifying 2z2 + -4z2 + 2z + 1 = 0 Reorder the terms: 1 + 2z + 2z2 + -4z2 = 0 Combine like terms: 2z2 + -4z2 = -2z2 1 + 2z + -2z2 = 0 Solving 1 + 2z + -2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -1z + z2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1z + 0.5 + z2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1z + z2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1z + z2 = 0 + 0.5 -1z + z2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1z + z2 = 0.5 The z term is -1z. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1z + 0.25 + z2 = 0.5 + 0.25 Reorder the terms: 0.25 + -1z + z2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + -1z + z2 = 0.75 Factor a perfect square on the left side: (z + -0.5)(z + -0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (z + -0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
z + -0.5 = 0.866025404 Simplifying z + -0.5 = 0.866025404 Reorder the terms: -0.5 + z = 0.866025404 Solving -0.5 + z = 0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 0.866025404 + 0.5 z = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 z = 1.366025404 Simplifying z = 1.366025404Subproblem 2
z + -0.5 = -0.866025404 Simplifying z + -0.5 = -0.866025404 Reorder the terms: -0.5 + z = -0.866025404 Solving -0.5 + z = -0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -0.866025404 + 0.5 z = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 z = -0.366025404 Simplifying z = -0.366025404Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.366025404, -0.366025404}
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