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Simplifying z2 + -7z = 1 Reorder the terms: -7z + z2 = 1 Solving -7z + z2 = 1 Solving for variable 'z'. Reorder the terms: -1 + -7z + z2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -7z + z2 = 0 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 = 13.25 12.25 + -7z + z2 = 13.25 Factor a perfect square on the left side: (z + -3.5)(z + -3.5) = 13.25 Calculate the square root of the right side: 3.640054945 Break this problem into two subproblems by setting (z + -3.5) equal to 3.640054945 and -3.640054945.Subproblem 1
z + -3.5 = 3.640054945 Simplifying z + -3.5 = 3.640054945 Reorder the terms: -3.5 + z = 3.640054945 Solving -3.5 + z = 3.640054945 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 = 3.640054945 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = 3.640054945 + 3.5 z = 3.640054945 + 3.5 Combine like terms: 3.640054945 + 3.5 = 7.140054945 z = 7.140054945 Simplifying z = 7.140054945Subproblem 2
z + -3.5 = -3.640054945 Simplifying z + -3.5 = -3.640054945 Reorder the terms: -3.5 + z = -3.640054945 Solving -3.5 + z = -3.640054945 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 = -3.640054945 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = -3.640054945 + 3.5 z = -3.640054945 + 3.5 Combine like terms: -3.640054945 + 3.5 = -0.140054945 z = -0.140054945 Simplifying z = -0.140054945Solution
The solution to the problem is based on the solutions from the subproblems. z = {7.140054945, -0.140054945}
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