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Simplifying 18z2 + 37z + 16 = 0 Reorder the terms: 16 + 37z + 18z2 = 0 Solving 16 + 37z + 18z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.8888888889 + 2.055555556z + z2 = 0 Move the constant term to the right: Add '-0.8888888889' to each side of the equation. 0.8888888889 + 2.055555556z + -0.8888888889 + z2 = 0 + -0.8888888889 Reorder the terms: 0.8888888889 + -0.8888888889 + 2.055555556z + z2 = 0 + -0.8888888889 Combine like terms: 0.8888888889 + -0.8888888889 = 0.0000000000 0.0000000000 + 2.055555556z + z2 = 0 + -0.8888888889 2.055555556z + z2 = 0 + -0.8888888889 Combine like terms: 0 + -0.8888888889 = -0.8888888889 2.055555556z + z2 = -0.8888888889 The z term is 2.055555556z. Take half its coefficient (1.027777778). Square it (1.056327161) and add it to both sides. Add '1.056327161' to each side of the equation. 2.055555556z + 1.056327161 + z2 = -0.8888888889 + 1.056327161 Reorder the terms: 1.056327161 + 2.055555556z + z2 = -0.8888888889 + 1.056327161 Combine like terms: -0.8888888889 + 1.056327161 = 0.1674382721 1.056327161 + 2.055555556z + z2 = 0.1674382721 Factor a perfect square on the left side: (z + 1.027777778)(z + 1.027777778) = 0.1674382721 Calculate the square root of the right side: 0.409192219 Break this problem into two subproblems by setting (z + 1.027777778) equal to 0.409192219 and -0.409192219.Subproblem 1
z + 1.027777778 = 0.409192219 Simplifying z + 1.027777778 = 0.409192219 Reorder the terms: 1.027777778 + z = 0.409192219 Solving 1.027777778 + z = 0.409192219 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.027777778' to each side of the equation. 1.027777778 + -1.027777778 + z = 0.409192219 + -1.027777778 Combine like terms: 1.027777778 + -1.027777778 = 0.000000000 0.000000000 + z = 0.409192219 + -1.027777778 z = 0.409192219 + -1.027777778 Combine like terms: 0.409192219 + -1.027777778 = -0.618585559 z = -0.618585559 Simplifying z = -0.618585559Subproblem 2
z + 1.027777778 = -0.409192219 Simplifying z + 1.027777778 = -0.409192219 Reorder the terms: 1.027777778 + z = -0.409192219 Solving 1.027777778 + z = -0.409192219 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.027777778' to each side of the equation. 1.027777778 + -1.027777778 + z = -0.409192219 + -1.027777778 Combine like terms: 1.027777778 + -1.027777778 = 0.000000000 0.000000000 + z = -0.409192219 + -1.027777778 z = -0.409192219 + -1.027777778 Combine like terms: -0.409192219 + -1.027777778 = -1.436969997 z = -1.436969997 Simplifying z = -1.436969997Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.618585559, -1.436969997}
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