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Simplifying 1 + 4z + z2 = 0 Solving 1 + 4z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + 4z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4z + z2 = 0 + -1 4z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 4z + z2 = -1 The z term is 4z. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4z + 4 + z2 = -1 + 4 Reorder the terms: 4 + 4z + z2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + 4z + z2 = 3 Factor a perfect square on the left side: (z + 2)(z + 2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (z + 2) equal to 1.732050808 and -1.732050808.Subproblem 1
z + 2 = 1.732050808 Simplifying z + 2 = 1.732050808 Reorder the terms: 2 + z = 1.732050808 Solving 2 + z = 1.732050808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + z = 1.732050808 + -2 z = 1.732050808 + -2 Combine like terms: 1.732050808 + -2 = -0.267949192 z = -0.267949192 Simplifying z = -0.267949192Subproblem 2
z + 2 = -1.732050808 Simplifying z + 2 = -1.732050808 Reorder the terms: 2 + z = -1.732050808 Solving 2 + z = -1.732050808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + z = -1.732050808 + -2 z = -1.732050808 + -2 Combine like terms: -1.732050808 + -2 = -3.732050808 z = -3.732050808 Simplifying z = -3.732050808Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.267949192, -3.732050808}
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