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Simplifying z3 + 5z2 + -18z = 0 Reorder the terms: -18z + 5z2 + z3 = 0 Solving -18z + 5z2 + z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(-18 + 5z + 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 '(-18 + 5z + z2)' equal to zero and attempt to solve: Simplifying -18 + 5z + z2 = 0 Solving -18 + 5z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 5z + 18 + z2 = 0 + 18 Reorder the terms: -18 + 18 + 5z + z2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 5z + z2 = 0 + 18 5z + z2 = 0 + 18 Combine like terms: 0 + 18 = 18 5z + z2 = 18 The z term is 5z. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5z + 6.25 + z2 = 18 + 6.25 Reorder the terms: 6.25 + 5z + z2 = 18 + 6.25 Combine like terms: 18 + 6.25 = 24.25 6.25 + 5z + z2 = 24.25 Factor a perfect square on the left side: (z + 2.5)(z + 2.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (z + 2.5) equal to 4.924428901 and -4.924428901.Subproblem 1
z + 2.5 = 4.924428901 Simplifying z + 2.5 = 4.924428901 Reorder the terms: 2.5 + z = 4.924428901 Solving 2.5 + z = 4.924428901 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 4.924428901 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 4.924428901 + -2.5 z = 4.924428901 + -2.5 Combine like terms: 4.924428901 + -2.5 = 2.424428901 z = 2.424428901 Simplifying z = 2.424428901Subproblem 2
z + 2.5 = -4.924428901 Simplifying z + 2.5 = -4.924428901 Reorder the terms: 2.5 + z = -4.924428901 Solving 2.5 + z = -4.924428901 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -4.924428901 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -4.924428901 + -2.5 z = -4.924428901 + -2.5 Combine like terms: -4.924428901 + -2.5 = -7.424428901 z = -7.424428901 Simplifying z = -7.424428901Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.424428901, -7.424428901}Solution
z = {0, 2.424428901, -7.424428901}
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