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Simplifying z2 + 16z = -47 Reorder the terms: 16z + z2 = -47 Solving 16z + z2 = -47 Solving for variable 'z'. Reorder the terms: 47 + 16z + z2 = -47 + 47 Combine like terms: -47 + 47 = 0 47 + 16z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-47' to each side of the equation. 47 + 16z + -47 + z2 = 0 + -47 Reorder the terms: 47 + -47 + 16z + z2 = 0 + -47 Combine like terms: 47 + -47 = 0 0 + 16z + z2 = 0 + -47 16z + z2 = 0 + -47 Combine like terms: 0 + -47 = -47 16z + z2 = -47 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 = -47 + 64 Reorder the terms: 64 + 16z + z2 = -47 + 64 Combine like terms: -47 + 64 = 17 64 + 16z + z2 = 17 Factor a perfect square on the left side: (z + 8)(z + 8) = 17 Calculate the square root of the right side: 4.123105626 Break this problem into two subproblems by setting (z + 8) equal to 4.123105626 and -4.123105626.Subproblem 1
z + 8 = 4.123105626 Simplifying z + 8 = 4.123105626 Reorder the terms: 8 + z = 4.123105626 Solving 8 + z = 4.123105626 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 = 4.123105626 + -8 Combine like terms: 8 + -8 = 0 0 + z = 4.123105626 + -8 z = 4.123105626 + -8 Combine like terms: 4.123105626 + -8 = -3.876894374 z = -3.876894374 Simplifying z = -3.876894374Subproblem 2
z + 8 = -4.123105626 Simplifying z + 8 = -4.123105626 Reorder the terms: 8 + z = -4.123105626 Solving 8 + z = -4.123105626 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 = -4.123105626 + -8 Combine like terms: 8 + -8 = 0 0 + z = -4.123105626 + -8 z = -4.123105626 + -8 Combine like terms: -4.123105626 + -8 = -12.123105626 z = -12.123105626 Simplifying z = -12.123105626Solution
The solution to the problem is based on the solutions from the subproblems. z = {-3.876894374, -12.123105626}
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