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Simplifying z2 + 16z + 64 = 7 Reorder the terms: 64 + 16z + z2 = 7 Solving 64 + 16z + z2 = 7 Solving for variable 'z'. Reorder the terms: 64 + -7 + 16z + z2 = 7 + -7 Combine like terms: 64 + -7 = 57 57 + 16z + z2 = 7 + -7 Combine like terms: 7 + -7 = 0 57 + 16z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-57' to each side of the equation. 57 + 16z + -57 + z2 = 0 + -57 Reorder the terms: 57 + -57 + 16z + z2 = 0 + -57 Combine like terms: 57 + -57 = 0 0 + 16z + z2 = 0 + -57 16z + z2 = 0 + -57 Combine like terms: 0 + -57 = -57 16z + z2 = -57 The z term is 16z. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16z + 64 + z2 = -57 + 64 Reorder the terms: 64 + 16z + z2 = -57 + 64 Combine like terms: -57 + 64 = 7 64 + 16z + z2 = 7 Factor a perfect square on the left side: (z + 8)(z + 8) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (z + 8) equal to 2.645751311 and -2.645751311.Subproblem 1
z + 8 = 2.645751311 Simplifying z + 8 = 2.645751311 Reorder the terms: 8 + z = 2.645751311 Solving 8 + z = 2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + z = 2.645751311 + -8 Combine like terms: 8 + -8 = 0 0 + z = 2.645751311 + -8 z = 2.645751311 + -8 Combine like terms: 2.645751311 + -8 = -5.354248689 z = -5.354248689 Simplifying z = -5.354248689Subproblem 2
z + 8 = -2.645751311 Simplifying z + 8 = -2.645751311 Reorder the terms: 8 + z = -2.645751311 Solving 8 + z = -2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + z = -2.645751311 + -8 Combine like terms: 8 + -8 = 0 0 + z = -2.645751311 + -8 z = -2.645751311 + -8 Combine like terms: -2.645751311 + -8 = -10.645751311 z = -10.645751311 Simplifying z = -10.645751311Solution
The solution to the problem is based on the solutions from the subproblems. z = {-5.354248689, -10.645751311}
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