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Simplifying z2 + 7z = 5 Reorder the terms: 7z + z2 = 5 Solving 7z + z2 = 5 Solving for variable 'z'. Reorder the terms: -5 + 7z + z2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 7z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 7z + 5 + z2 = 0 + 5 Reorder the terms: -5 + 5 + 7z + z2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 7z + z2 = 0 + 5 7z + z2 = 0 + 5 Combine like terms: 0 + 5 = 5 7z + z2 = 5 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 = 5 + 12.25 Reorder the terms: 12.25 + 7z + z2 = 5 + 12.25 Combine like terms: 5 + 12.25 = 17.25 12.25 + 7z + z2 = 17.25 Factor a perfect square on the left side: (z + 3.5)(z + 3.5) = 17.25 Calculate the square root of the right side: 4.153311931 Break this problem into two subproblems by setting (z + 3.5) equal to 4.153311931 and -4.153311931.Subproblem 1
z + 3.5 = 4.153311931 Simplifying z + 3.5 = 4.153311931 Reorder the terms: 3.5 + z = 4.153311931 Solving 3.5 + z = 4.153311931 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 = 4.153311931 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 4.153311931 + -3.5 z = 4.153311931 + -3.5 Combine like terms: 4.153311931 + -3.5 = 0.653311931 z = 0.653311931 Simplifying z = 0.653311931Subproblem 2
z + 3.5 = -4.153311931 Simplifying z + 3.5 = -4.153311931 Reorder the terms: 3.5 + z = -4.153311931 Solving 3.5 + z = -4.153311931 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 = -4.153311931 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -4.153311931 + -3.5 z = -4.153311931 + -3.5 Combine like terms: -4.153311931 + -3.5 = -7.653311931 z = -7.653311931 Simplifying z = -7.653311931Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.653311931, -7.653311931}
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