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Simplifying 2z3 + 16z2 + -4z = 0 Reorder the terms: -4z + 16z2 + 2z3 = 0 Solving -4z + 16z2 + 2z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2z'. 2z(-2 + 8z + z2) = 0 Ignore the factor 2.Subproblem 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 '(-2 + 8z + z2)' equal to zero and attempt to solve: Simplifying -2 + 8z + z2 = 0 Solving -2 + 8z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 8z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 8z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 8z + z2 = 0 + 2 8z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 8z + z2 = 2 The z term is 8z. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8z + 16 + z2 = 2 + 16 Reorder the terms: 16 + 8z + z2 = 2 + 16 Combine like terms: 2 + 16 = 18 16 + 8z + z2 = 18 Factor a perfect square on the left side: (z + 4)(z + 4) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (z + 4) equal to 4.242640687 and -4.242640687.Subproblem 1
z + 4 = 4.242640687 Simplifying z + 4 = 4.242640687 Reorder the terms: 4 + z = 4.242640687 Solving 4 + z = 4.242640687 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + z = 4.242640687 + -4 Combine like terms: 4 + -4 = 0 0 + z = 4.242640687 + -4 z = 4.242640687 + -4 Combine like terms: 4.242640687 + -4 = 0.242640687 z = 0.242640687 Simplifying z = 0.242640687Subproblem 2
z + 4 = -4.242640687 Simplifying z + 4 = -4.242640687 Reorder the terms: 4 + z = -4.242640687 Solving 4 + z = -4.242640687 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + z = -4.242640687 + -4 Combine like terms: 4 + -4 = 0 0 + z = -4.242640687 + -4 z = -4.242640687 + -4 Combine like terms: -4.242640687 + -4 = -8.242640687 z = -8.242640687 Simplifying z = -8.242640687Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.242640687, -8.242640687}Solution
z = {0, 0.242640687, -8.242640687}
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