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Simplifying 2z3 + -12z2 + 3z = 0 Reorder the terms: 3z + -12z2 + 2z3 = 0 Solving 3z + -12z2 + 2z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(3 + -12z + 2z2) = 0Subproblem 1
Set the factor 'z' equal to zero and attempt to solve: Simplifying z = 0 Solving z = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z = 0Subproblem 2
Set the factor '(3 + -12z + 2z2)' equal to zero and attempt to solve: Simplifying 3 + -12z + 2z2 = 0 Solving 3 + -12z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -6z + z2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -6z + -1.5 + z2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -6z + z2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -6z + z2 = 0 + -1.5 -6z + z2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -6z + z2 = -1.5 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = -1.5 + 9 Reorder the terms: 9 + -6z + z2 = -1.5 + 9 Combine like terms: -1.5 + 9 = 7.5 9 + -6z + z2 = 7.5 Factor a perfect square on the left side: (z + -3)(z + -3) = 7.5 Calculate the square root of the right side: 2.738612788 Break this problem into two subproblems by setting (z + -3) equal to 2.738612788 and -2.738612788.Subproblem 1
z + -3 = 2.738612788 Simplifying z + -3 = 2.738612788 Reorder the terms: -3 + z = 2.738612788 Solving -3 + z = 2.738612788 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 2.738612788 + 3 Combine like terms: -3 + 3 = 0 0 + z = 2.738612788 + 3 z = 2.738612788 + 3 Combine like terms: 2.738612788 + 3 = 5.738612788 z = 5.738612788 Simplifying z = 5.738612788Subproblem 2
z + -3 = -2.738612788 Simplifying z + -3 = -2.738612788 Reorder the terms: -3 + z = -2.738612788 Solving -3 + z = -2.738612788 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -2.738612788 + 3 Combine like terms: -3 + 3 = 0 0 + z = -2.738612788 + 3 z = -2.738612788 + 3 Combine like terms: -2.738612788 + 3 = 0.261387212 z = 0.261387212 Simplifying z = 0.261387212Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.738612788, 0.261387212}Solution
z = {0, 5.738612788, 0.261387212}
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