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Simplifying z * z + 16z + 41 = 0 Multiply z * z z^{2}+ 16z + 41 = 0 Reorder the terms: 41 + 16z + z^{2}= 0 Solving 41 + 16z + z^{2}= 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-41' to each side of the equation. 41 + 16z + -41 + z^{2}= 0 + -41 Reorder the terms: 41 + -41 + 16z + z^{2}= 0 + -41 Combine like terms: 41 + -41 = 0 0 + 16z + z^{2}= 0 + -41 16z + z^{2}= 0 + -41 Combine like terms: 0 + -41 = -41 16z + z^{2}= -41 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 + z^{2}= -41 + 64 Reorder the terms: 64 + 16z + z^{2}= -41 + 64 Combine like terms: -41 + 64 = 23 64 + 16z + z^{2}= 23 Factor a perfect square on the left side: (z + 8)(z + 8) = 23 Calculate the square root of the right side: 4.795831523 Break this problem into two subproblems by setting (z + 8) equal to 4.795831523 and -4.795831523.## Subproblem 1

z + 8 = 4.795831523 Simplifying z + 8 = 4.795831523 Reorder the terms: 8 + z = 4.795831523 Solving 8 + z = 4.795831523 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 = 4.795831523 + -8 Combine like terms: 8 + -8 = 0 0 + z = 4.795831523 + -8 z = 4.795831523 + -8 Combine like terms: 4.795831523 + -8 = -3.204168477 z = -3.204168477 Simplifying z = -3.204168477## Subproblem 2

z + 8 = -4.795831523 Simplifying z + 8 = -4.795831523 Reorder the terms: 8 + z = -4.795831523 Solving 8 + z = -4.795831523 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 = -4.795831523 + -8 Combine like terms: 8 + -8 = 0 0 + z = -4.795831523 + -8 z = -4.795831523 + -8 Combine like terms: -4.795831523 + -8 = -12.795831523 z = -12.795831523 Simplifying z = -12.795831523## Solution

The solution to the problem is based on the solutions from the subproblems. z = {-3.204168477, -12.795831523}

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