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Simplifying z3 + -7z2 + -12z = 0 Reorder the terms: -12z + -7z2 + z3 = 0 Solving -12z + -7z2 + z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(-12 + -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 '(-12 + -7z + z2)' equal to zero and attempt to solve: Simplifying -12 + -7z + z2 = 0 Solving -12 + -7z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -7z + 12 + z2 = 0 + 12 Reorder the terms: -12 + 12 + -7z + z2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -7z + z2 = 0 + 12 -7z + z2 = 0 + 12 Combine like terms: 0 + 12 = 12 -7z + z2 = 12 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 = 12 + 12.25 Reorder the terms: 12.25 + -7z + z2 = 12 + 12.25 Combine like terms: 12 + 12.25 = 24.25 12.25 + -7z + z2 = 24.25 Factor a perfect square on the left side: (z + -3.5)(z + -3.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (z + -3.5) equal to 4.924428901 and -4.924428901.Subproblem 1
z + -3.5 = 4.924428901 Simplifying z + -3.5 = 4.924428901 Reorder the terms: -3.5 + z = 4.924428901 Solving -3.5 + z = 4.924428901 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 = 4.924428901 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = 4.924428901 + 3.5 z = 4.924428901 + 3.5 Combine like terms: 4.924428901 + 3.5 = 8.424428901 z = 8.424428901 Simplifying z = 8.424428901Subproblem 2
z + -3.5 = -4.924428901 Simplifying z + -3.5 = -4.924428901 Reorder the terms: -3.5 + z = -4.924428901 Solving -3.5 + z = -4.924428901 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 = -4.924428901 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = -4.924428901 + 3.5 z = -4.924428901 + 3.5 Combine like terms: -4.924428901 + 3.5 = -1.424428901 z = -1.424428901 Simplifying z = -1.424428901Solution
The solution to the problem is based on the solutions from the subproblems. z = {8.424428901, -1.424428901}Solution
z = {0, 8.424428901, -1.424428901}
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