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Simplifying (z2) + 12z + 19 = 0 z2 + 12z + 19 = 0 Reorder the terms: 19 + 12z + z2 = 0 Solving 19 + 12z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + 12z + -19 + z2 = 0 + -19 Reorder the terms: 19 + -19 + 12z + z2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + 12z + z2 = 0 + -19 12z + z2 = 0 + -19 Combine like terms: 0 + -19 = -19 12z + z2 = -19 The z term is 12z. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12z + 36 + z2 = -19 + 36 Reorder the terms: 36 + 12z + z2 = -19 + 36 Combine like terms: -19 + 36 = 17 36 + 12z + z2 = 17 Factor a perfect square on the left side: (z + 6)(z + 6) = 17 Calculate the square root of the right side: 4.123105626 Break this problem into two subproblems by setting (z + 6) equal to 4.123105626 and -4.123105626.Subproblem 1
z + 6 = 4.123105626 Simplifying z + 6 = 4.123105626 Reorder the terms: 6 + z = 4.123105626 Solving 6 + z = 4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = 4.123105626 + -6 Combine like terms: 6 + -6 = 0 0 + z = 4.123105626 + -6 z = 4.123105626 + -6 Combine like terms: 4.123105626 + -6 = -1.876894374 z = -1.876894374 Simplifying z = -1.876894374Subproblem 2
z + 6 = -4.123105626 Simplifying z + 6 = -4.123105626 Reorder the terms: 6 + z = -4.123105626 Solving 6 + z = -4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = -4.123105626 + -6 Combine like terms: 6 + -6 = 0 0 + z = -4.123105626 + -6 z = -4.123105626 + -6 Combine like terms: -4.123105626 + -6 = -10.123105626 z = -10.123105626 Simplifying z = -10.123105626Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.876894374, -10.123105626}
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