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Simplifying 36z2 + 32z + 5 = 0 Reorder the terms: 5 + 32z + 36z2 = 0 Solving 5 + 32z + 36z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. 0.1388888889 + 0.8888888889z + z2 = 0 Move the constant term to the right: Add '-0.1388888889' to each side of the equation. 0.1388888889 + 0.8888888889z + -0.1388888889 + z2 = 0 + -0.1388888889 Reorder the terms: 0.1388888889 + -0.1388888889 + 0.8888888889z + z2 = 0 + -0.1388888889 Combine like terms: 0.1388888889 + -0.1388888889 = 0.0000000000 0.0000000000 + 0.8888888889z + z2 = 0 + -0.1388888889 0.8888888889z + z2 = 0 + -0.1388888889 Combine like terms: 0 + -0.1388888889 = -0.1388888889 0.8888888889z + z2 = -0.1388888889 The z term is 0.8888888889z. Take half its coefficient (0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. 0.8888888889z + 0.1975308642 + z2 = -0.1388888889 + 0.1975308642 Reorder the terms: 0.1975308642 + 0.8888888889z + z2 = -0.1388888889 + 0.1975308642 Combine like terms: -0.1388888889 + 0.1975308642 = 0.0586419753 0.1975308642 + 0.8888888889z + z2 = 0.0586419753 Factor a perfect square on the left side: (z + 0.4444444445)(z + 0.4444444445) = 0.0586419753 Calculate the square root of the right side: 0.242161052 Break this problem into two subproblems by setting (z + 0.4444444445) equal to 0.242161052 and -0.242161052.Subproblem 1
z + 0.4444444445 = 0.242161052 Simplifying z + 0.4444444445 = 0.242161052 Reorder the terms: 0.4444444445 + z = 0.242161052 Solving 0.4444444445 + z = 0.242161052 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + z = 0.242161052 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + z = 0.242161052 + -0.4444444445 z = 0.242161052 + -0.4444444445 Combine like terms: 0.242161052 + -0.4444444445 = -0.2022833925 z = -0.2022833925 Simplifying z = -0.2022833925Subproblem 2
z + 0.4444444445 = -0.242161052 Simplifying z + 0.4444444445 = -0.242161052 Reorder the terms: 0.4444444445 + z = -0.242161052 Solving 0.4444444445 + z = -0.242161052 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + z = -0.242161052 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + z = -0.242161052 + -0.4444444445 z = -0.242161052 + -0.4444444445 Combine like terms: -0.242161052 + -0.4444444445 = -0.6866054965 z = -0.6866054965 Simplifying z = -0.6866054965Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.2022833925, -0.6866054965}
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