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Simplifying z2 + 7z + 1 = 0 Reorder the terms: 1 + 7z + z2 = 0 Solving 1 + 7z + 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 + 7z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + 7z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 7z + z2 = 0 + -1 7z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 7z + z2 = -1 The z term is 7z. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7z + 12.25 + z2 = -1 + 12.25 Reorder the terms: 12.25 + 7z + z2 = -1 + 12.25 Combine like terms: -1 + 12.25 = 11.25 12.25 + 7z + z2 = 11.25 Factor a perfect square on the left side: (z + 3.5)(z + 3.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (z + 3.5) equal to 3.354101966 and -3.354101966.Subproblem 1
z + 3.5 = 3.354101966 Simplifying z + 3.5 = 3.354101966 Reorder the terms: 3.5 + z = 3.354101966 Solving 3.5 + z = 3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 3.354101966 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 3.354101966 + -3.5 z = 3.354101966 + -3.5 Combine like terms: 3.354101966 + -3.5 = -0.145898034 z = -0.145898034 Simplifying z = -0.145898034Subproblem 2
z + 3.5 = -3.354101966 Simplifying z + 3.5 = -3.354101966 Reorder the terms: 3.5 + z = -3.354101966 Solving 3.5 + z = -3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -3.354101966 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -3.354101966 + -3.5 z = -3.354101966 + -3.5 Combine like terms: -3.354101966 + -3.5 = -6.854101966 z = -6.854101966 Simplifying z = -6.854101966Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.145898034, -6.854101966}
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