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Simplifying 2z2 + -18z + -24 = 0 Reorder the terms: -24 + -18z + 2z2 = 0 Solving -24 + -18z + 2z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + -9z + z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + -9z + z2)' equal to zero and attempt to solve: Simplifying -12 + -9z + z2 = 0 Solving -12 + -9z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -9z + 12 + z2 = 0 + 12 Reorder the terms: -12 + 12 + -9z + z2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -9z + z2 = 0 + 12 -9z + z2 = 0 + 12 Combine like terms: 0 + 12 = 12 -9z + z2 = 12 The z term is -9z. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9z + 20.25 + z2 = 12 + 20.25 Reorder the terms: 20.25 + -9z + z2 = 12 + 20.25 Combine like terms: 12 + 20.25 = 32.25 20.25 + -9z + z2 = 32.25 Factor a perfect square on the left side: (z + -4.5)(z + -4.5) = 32.25 Calculate the square root of the right side: 5.678908346 Break this problem into two subproblems by setting (z + -4.5) equal to 5.678908346 and -5.678908346.Subproblem 1
z + -4.5 = 5.678908346 Simplifying z + -4.5 = 5.678908346 Reorder the terms: -4.5 + z = 5.678908346 Solving -4.5 + z = 5.678908346 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = 5.678908346 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = 5.678908346 + 4.5 z = 5.678908346 + 4.5 Combine like terms: 5.678908346 + 4.5 = 10.178908346 z = 10.178908346 Simplifying z = 10.178908346Subproblem 2
z + -4.5 = -5.678908346 Simplifying z + -4.5 = -5.678908346 Reorder the terms: -4.5 + z = -5.678908346 Solving -4.5 + z = -5.678908346 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = -5.678908346 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = -5.678908346 + 4.5 z = -5.678908346 + 4.5 Combine like terms: -5.678908346 + 4.5 = -1.178908346 z = -1.178908346 Simplifying z = -1.178908346Solution
The solution to the problem is based on the solutions from the subproblems. z = {10.178908346, -1.178908346}Solution
z = {10.178908346, -1.178908346}
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