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Simplifying 36z2 + 96z + 5 = 0 Reorder the terms: 5 + 96z + 36z2 = 0 Solving 5 + 96z + 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 + 2.666666667z + z2 = 0 Move the constant term to the right: Add '-0.1388888889' to each side of the equation. 0.1388888889 + 2.666666667z + -0.1388888889 + z2 = 0 + -0.1388888889 Reorder the terms: 0.1388888889 + -0.1388888889 + 2.666666667z + z2 = 0 + -0.1388888889 Combine like terms: 0.1388888889 + -0.1388888889 = 0.0000000000 0.0000000000 + 2.666666667z + z2 = 0 + -0.1388888889 2.666666667z + z2 = 0 + -0.1388888889 Combine like terms: 0 + -0.1388888889 = -0.1388888889 2.666666667z + z2 = -0.1388888889 The z term is 2.666666667z. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667z + 1.777777780 + z2 = -0.1388888889 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667z + z2 = -0.1388888889 + 1.777777780 Combine like terms: -0.1388888889 + 1.777777780 = 1.6388888911 1.777777780 + 2.666666667z + z2 = 1.6388888911 Factor a perfect square on the left side: (z + 1.333333334)(z + 1.333333334) = 1.6388888911 Calculate the square root of the right side: 1.280190959 Break this problem into two subproblems by setting (z + 1.333333334) equal to 1.280190959 and -1.280190959.Subproblem 1
z + 1.333333334 = 1.280190959 Simplifying z + 1.333333334 = 1.280190959 Reorder the terms: 1.333333334 + z = 1.280190959 Solving 1.333333334 + z = 1.280190959 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + z = 1.280190959 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + z = 1.280190959 + -1.333333334 z = 1.280190959 + -1.333333334 Combine like terms: 1.280190959 + -1.333333334 = -0.053142375 z = -0.053142375 Simplifying z = -0.053142375Subproblem 2
z + 1.333333334 = -1.280190959 Simplifying z + 1.333333334 = -1.280190959 Reorder the terms: 1.333333334 + z = -1.280190959 Solving 1.333333334 + z = -1.280190959 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + z = -1.280190959 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + z = -1.280190959 + -1.333333334 z = -1.280190959 + -1.333333334 Combine like terms: -1.280190959 + -1.333333334 = -2.613524293 z = -2.613524293 Simplifying z = -2.613524293Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.053142375, -2.613524293}
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