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Simplifying (z2) + 5z + -7 = 0 z2 + 5z + -7 = 0 Reorder the terms: -7 + 5z + z2 = 0 Solving -7 + 5z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + 5z + 7 + z2 = 0 + 7 Reorder the terms: -7 + 7 + 5z + z2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 5z + z2 = 0 + 7 5z + z2 = 0 + 7 Combine like terms: 0 + 7 = 7 5z + z2 = 7 The z term is 5z. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5z + 6.25 + z2 = 7 + 6.25 Reorder the terms: 6.25 + 5z + z2 = 7 + 6.25 Combine like terms: 7 + 6.25 = 13.25 6.25 + 5z + z2 = 13.25 Factor a perfect square on the left side: (z + 2.5)(z + 2.5) = 13.25 Calculate the square root of the right side: 3.640054945 Break this problem into two subproblems by setting (z + 2.5) equal to 3.640054945 and -3.640054945.Subproblem 1
z + 2.5 = 3.640054945 Simplifying z + 2.5 = 3.640054945 Reorder the terms: 2.5 + z = 3.640054945 Solving 2.5 + z = 3.640054945 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 3.640054945 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 3.640054945 + -2.5 z = 3.640054945 + -2.5 Combine like terms: 3.640054945 + -2.5 = 1.140054945 z = 1.140054945 Simplifying z = 1.140054945Subproblem 2
z + 2.5 = -3.640054945 Simplifying z + 2.5 = -3.640054945 Reorder the terms: 2.5 + z = -3.640054945 Solving 2.5 + z = -3.640054945 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -3.640054945 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -3.640054945 + -2.5 z = -3.640054945 + -2.5 Combine like terms: -3.640054945 + -2.5 = -6.140054945 z = -6.140054945 Simplifying z = -6.140054945Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.140054945, -6.140054945}
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