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Simplifying 7z2 + 8z + z3 = 0 Reorder the terms: 8z + 7z2 + z3 = 0 Solving 8z + 7z2 + z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(8 + 7z + z2) = 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 '(8 + 7z + z2)' equal to zero and attempt to solve: Simplifying 8 + 7z + z2 = 0 Solving 8 + 7z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + 7z + -8 + z2 = 0 + -8 Reorder the terms: 8 + -8 + 7z + z2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + 7z + z2 = 0 + -8 7z + z2 = 0 + -8 Combine like terms: 0 + -8 = -8 7z + z2 = -8 The z term is 7z. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7z + 12.25 + z2 = -8 + 12.25 Reorder the terms: 12.25 + 7z + z2 = -8 + 12.25 Combine like terms: -8 + 12.25 = 4.25 12.25 + 7z + z2 = 4.25 Factor a perfect square on the left side: (z + 3.5)(z + 3.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (z + 3.5) equal to 2.061552813 and -2.061552813.Subproblem 1
z + 3.5 = 2.061552813 Simplifying z + 3.5 = 2.061552813 Reorder the terms: 3.5 + z = 2.061552813 Solving 3.5 + z = 2.061552813 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 2.061552813 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 2.061552813 + -3.5 z = 2.061552813 + -3.5 Combine like terms: 2.061552813 + -3.5 = -1.438447187 z = -1.438447187 Simplifying z = -1.438447187Subproblem 2
z + 3.5 = -2.061552813 Simplifying z + 3.5 = -2.061552813 Reorder the terms: 3.5 + z = -2.061552813 Solving 3.5 + z = -2.061552813 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -2.061552813 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -2.061552813 + -3.5 z = -2.061552813 + -3.5 Combine like terms: -2.061552813 + -3.5 = -5.561552813 z = -5.561552813 Simplifying z = -5.561552813Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.438447187, -5.561552813}Solution
z = {0, -1.438447187, -5.561552813}
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