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Simplifying 2z3 + 8z2 + -4z = 0 Reorder the terms: -4z + 8z2 + 2z3 = 0 Solving -4z + 8z2 + 2z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2z'. 2z(-2 + 4z + 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 + 4z + z2)' equal to zero and attempt to solve: Simplifying -2 + 4z + z2 = 0 Solving -2 + 4z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 4z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 4z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 4z + z2 = 0 + 2 4z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 4z + z2 = 2 The z term is 4z. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4z + 4 + z2 = 2 + 4 Reorder the terms: 4 + 4z + z2 = 2 + 4 Combine like terms: 2 + 4 = 6 4 + 4z + z2 = 6 Factor a perfect square on the left side: (z + 2)(z + 2) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (z + 2) equal to 2.449489743 and -2.449489743.Subproblem 1
z + 2 = 2.449489743 Simplifying z + 2 = 2.449489743 Reorder the terms: 2 + z = 2.449489743 Solving 2 + z = 2.449489743 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 2.449489743 + -2 Combine like terms: 2 + -2 = 0 0 + z = 2.449489743 + -2 z = 2.449489743 + -2 Combine like terms: 2.449489743 + -2 = 0.449489743 z = 0.449489743 Simplifying z = 0.449489743Subproblem 2
z + 2 = -2.449489743 Simplifying z + 2 = -2.449489743 Reorder the terms: 2 + z = -2.449489743 Solving 2 + z = -2.449489743 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -2.449489743 + -2 Combine like terms: 2 + -2 = 0 0 + z = -2.449489743 + -2 z = -2.449489743 + -2 Combine like terms: -2.449489743 + -2 = -4.449489743 z = -4.449489743 Simplifying z = -4.449489743Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.449489743, -4.449489743}Solution
z = {0, 0.449489743, -4.449489743}
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