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Simplifying 8z2 + -68z = 32 Reorder the terms: -68z + 8z2 = 32 Solving -68z + 8z2 = 32 Solving for variable 'z'. Reorder the terms: -32 + -68z + 8z2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + -68z + 8z2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-8 + -17z + 2z2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-8 + -17z + 2z2)' equal to zero and attempt to solve: Simplifying -8 + -17z + 2z2 = 0 Solving -8 + -17z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4 + -8.5z + z2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -8.5z + 4 + z2 = 0 + 4 Reorder the terms: -4 + 4 + -8.5z + z2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -8.5z + z2 = 0 + 4 -8.5z + z2 = 0 + 4 Combine like terms: 0 + 4 = 4 -8.5z + z2 = 4 The z term is -8.5z. Take half its coefficient (-4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. -8.5z + 18.0625 + z2 = 4 + 18.0625 Reorder the terms: 18.0625 + -8.5z + z2 = 4 + 18.0625 Combine like terms: 4 + 18.0625 = 22.0625 18.0625 + -8.5z + z2 = 22.0625 Factor a perfect square on the left side: (z + -4.25)(z + -4.25) = 22.0625 Calculate the square root of the right side: 4.697073557 Break this problem into two subproblems by setting (z + -4.25) equal to 4.697073557 and -4.697073557.Subproblem 1
z + -4.25 = 4.697073557 Simplifying z + -4.25 = 4.697073557 Reorder the terms: -4.25 + z = 4.697073557 Solving -4.25 + z = 4.697073557 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.25' to each side of the equation. -4.25 + 4.25 + z = 4.697073557 + 4.25 Combine like terms: -4.25 + 4.25 = 0.00 0.00 + z = 4.697073557 + 4.25 z = 4.697073557 + 4.25 Combine like terms: 4.697073557 + 4.25 = 8.947073557 z = 8.947073557 Simplifying z = 8.947073557Subproblem 2
z + -4.25 = -4.697073557 Simplifying z + -4.25 = -4.697073557 Reorder the terms: -4.25 + z = -4.697073557 Solving -4.25 + z = -4.697073557 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.25' to each side of the equation. -4.25 + 4.25 + z = -4.697073557 + 4.25 Combine like terms: -4.25 + 4.25 = 0.00 0.00 + z = -4.697073557 + 4.25 z = -4.697073557 + 4.25 Combine like terms: -4.697073557 + 4.25 = -0.447073557 z = -0.447073557 Simplifying z = -0.447073557Solution
The solution to the problem is based on the solutions from the subproblems. z = {8.947073557, -0.447073557}Solution
z = {8.947073557, -0.447073557}
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