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Simplifying -4z2 + -32z = -4 Reorder the terms: -32z + -4z2 = -4 Solving -32z + -4z2 = -4 Solving for variable 'z'. Reorder the terms: 4 + -32z + -4z2 = -4 + 4 Combine like terms: -4 + 4 = 0 4 + -32z + -4z2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1 + -8z + -1z2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + -8z + -1z2)' equal to zero and attempt to solve: Simplifying 1 + -8z + -1z2 = 0 Solving 1 + -8z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + 8z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 8z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + 8z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 8z + z2 = 0 + 1 8z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 8z + z2 = 1 The z term is 8z. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8z + 16 + z2 = 1 + 16 Reorder the terms: 16 + 8z + z2 = 1 + 16 Combine like terms: 1 + 16 = 17 16 + 8z + z2 = 17 Factor a perfect square on the left side: (z + 4)(z + 4) = 17 Calculate the square root of the right side: 4.123105626 Break this problem into two subproblems by setting (z + 4) equal to 4.123105626 and -4.123105626.Subproblem 1
z + 4 = 4.123105626 Simplifying z + 4 = 4.123105626 Reorder the terms: 4 + z = 4.123105626 Solving 4 + z = 4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + z = 4.123105626 + -4 Combine like terms: 4 + -4 = 0 0 + z = 4.123105626 + -4 z = 4.123105626 + -4 Combine like terms: 4.123105626 + -4 = 0.123105626 z = 0.123105626 Simplifying z = 0.123105626Subproblem 2
z + 4 = -4.123105626 Simplifying z + 4 = -4.123105626 Reorder the terms: 4 + z = -4.123105626 Solving 4 + z = -4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + z = -4.123105626 + -4 Combine like terms: 4 + -4 = 0 0 + z = -4.123105626 + -4 z = -4.123105626 + -4 Combine like terms: -4.123105626 + -4 = -8.123105626 z = -8.123105626 Simplifying z = -8.123105626Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.123105626, -8.123105626}Solution
z = {0.123105626, -8.123105626}
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