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Simplifying 3z3 + 9z2 + -6z = 0 Reorder the terms: -6z + 9z2 + 3z3 = 0 Solving -6z + 9z2 + 3z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '3z'. 3z(-2 + 3z + z2) = 0 Ignore the factor 3.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 + 3z + z2)' equal to zero and attempt to solve: Simplifying -2 + 3z + z2 = 0 Solving -2 + 3z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 3z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 3z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 3z + z2 = 0 + 2 3z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 3z + z2 = 2 The z term is 3z. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3z + 2.25 + z2 = 2 + 2.25 Reorder the terms: 2.25 + 3z + z2 = 2 + 2.25 Combine like terms: 2 + 2.25 = 4.25 2.25 + 3z + z2 = 4.25 Factor a perfect square on the left side: (z + 1.5)(z + 1.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (z + 1.5) equal to 2.061552813 and -2.061552813.Subproblem 1
z + 1.5 = 2.061552813 Simplifying z + 1.5 = 2.061552813 Reorder the terms: 1.5 + z = 2.061552813 Solving 1.5 + z = 2.061552813 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = 2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = 2.061552813 + -1.5 z = 2.061552813 + -1.5 Combine like terms: 2.061552813 + -1.5 = 0.561552813 z = 0.561552813 Simplifying z = 0.561552813Subproblem 2
z + 1.5 = -2.061552813 Simplifying z + 1.5 = -2.061552813 Reorder the terms: 1.5 + z = -2.061552813 Solving 1.5 + z = -2.061552813 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = -2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = -2.061552813 + -1.5 z = -2.061552813 + -1.5 Combine like terms: -2.061552813 + -1.5 = -3.561552813 z = -3.561552813 Simplifying z = -3.561552813Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.561552813, -3.561552813}Solution
z = {0, 0.561552813, -3.561552813}
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