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Simplifying 3z2 + -27z + 3 = 0 Reorder the terms: 3 + -27z + 3z2 = 0 Solving 3 + -27z + 3z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '3'. 3(1 + -9z + z2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(1 + -9z + z2)' equal to zero and attempt to solve: Simplifying 1 + -9z + z2 = 0 Solving 1 + -9z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -9z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + -9z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -9z + z2 = 0 + -1 -9z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 -9z + z2 = -1 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 = -1 + 20.25 Reorder the terms: 20.25 + -9z + z2 = -1 + 20.25 Combine like terms: -1 + 20.25 = 19.25 20.25 + -9z + z2 = 19.25 Factor a perfect square on the left side: (z + -4.5)(z + -4.5) = 19.25 Calculate the square root of the right side: 4.387482194 Break this problem into two subproblems by setting (z + -4.5) equal to 4.387482194 and -4.387482194.Subproblem 1
z + -4.5 = 4.387482194 Simplifying z + -4.5 = 4.387482194 Reorder the terms: -4.5 + z = 4.387482194 Solving -4.5 + z = 4.387482194 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 = 4.387482194 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = 4.387482194 + 4.5 z = 4.387482194 + 4.5 Combine like terms: 4.387482194 + 4.5 = 8.887482194 z = 8.887482194 Simplifying z = 8.887482194Subproblem 2
z + -4.5 = -4.387482194 Simplifying z + -4.5 = -4.387482194 Reorder the terms: -4.5 + z = -4.387482194 Solving -4.5 + z = -4.387482194 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 = -4.387482194 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = -4.387482194 + 4.5 z = -4.387482194 + 4.5 Combine like terms: -4.387482194 + 4.5 = 0.112517806 z = 0.112517806 Simplifying z = 0.112517806Solution
The solution to the problem is based on the solutions from the subproblems. z = {8.887482194, 0.112517806}Solution
z = {8.887482194, 0.112517806}
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