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Simplifying -4z2 + -64z + 52 = 0 Reorder the terms: 52 + -64z + -4z2 = 0 Solving 52 + -64z + -4z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '4'. 4(13 + -16z + -1z2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(13 + -16z + -1z2)' equal to zero and attempt to solve: Simplifying 13 + -16z + -1z2 = 0 Solving 13 + -16z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -13 + 16z + z2 = 0 Move the constant term to the right: Add '13' to each side of the equation. -13 + 16z + 13 + z2 = 0 + 13 Reorder the terms: -13 + 13 + 16z + z2 = 0 + 13 Combine like terms: -13 + 13 = 0 0 + 16z + z2 = 0 + 13 16z + z2 = 0 + 13 Combine like terms: 0 + 13 = 13 16z + z2 = 13 The z term is 16z. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16z + 64 + z2 = 13 + 64 Reorder the terms: 64 + 16z + z2 = 13 + 64 Combine like terms: 13 + 64 = 77 64 + 16z + z2 = 77 Factor a perfect square on the left side: (z + 8)(z + 8) = 77 Calculate the square root of the right side: 8.774964387 Break this problem into two subproblems by setting (z + 8) equal to 8.774964387 and -8.774964387.Subproblem 1
z + 8 = 8.774964387 Simplifying z + 8 = 8.774964387 Reorder the terms: 8 + z = 8.774964387 Solving 8 + z = 8.774964387 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + z = 8.774964387 + -8 Combine like terms: 8 + -8 = 0 0 + z = 8.774964387 + -8 z = 8.774964387 + -8 Combine like terms: 8.774964387 + -8 = 0.774964387 z = 0.774964387 Simplifying z = 0.774964387Subproblem 2
z + 8 = -8.774964387 Simplifying z + 8 = -8.774964387 Reorder the terms: 8 + z = -8.774964387 Solving 8 + z = -8.774964387 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + z = -8.774964387 + -8 Combine like terms: 8 + -8 = 0 0 + z = -8.774964387 + -8 z = -8.774964387 + -8 Combine like terms: -8.774964387 + -8 = -16.774964387 z = -16.774964387 Simplifying z = -16.774964387Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.774964387, -16.774964387}Solution
z = {0.774964387, -16.774964387}
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