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Simplifying -1z + -16z * 16z + -8 = 8 Reorder the terms for easier multiplication: -1z + -16 * 16z * z + -8 = 8 Multiply -16 * 16 -1z + -256z * z + -8 = 8 Multiply z * z -1z + -256z2 + -8 = 8 Reorder the terms: -8 + -1z + -256z2 = 8 Solving -8 + -1z + -256z2 = 8 Solving for variable 'z'. Reorder the terms: -8 + -8 + -1z + -256z2 = 8 + -8 Combine like terms: -8 + -8 = -16 -16 + -1z + -256z2 = 8 + -8 Combine like terms: 8 + -8 = 0 -16 + -1z + -256z2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(16 + z + 256z2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(16 + z + 256z2)' equal to zero and attempt to solve: Simplifying 16 + z + 256z2 = 0 Solving 16 + z + 256z2 = 0 Begin completing the square. Divide all terms by 256 the coefficient of the squared term: Divide each side by '256'. 0.0625 + 0.00390625z + z2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + 0.00390625z + -0.0625 + z2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + 0.00390625z + z2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + 0.00390625z + z2 = 0 + -0.0625 0.00390625z + z2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 0.00390625z + z2 = -0.0625 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.00390625z + 0.25 + z2 = -0.0625 + 0.25 Reorder the terms: 0.25 + 0.00390625z + z2 = -0.0625 + 0.25 Combine like terms: -0.0625 + 0.25 = 0.1875 0.25 + 0.00390625z + z2 = 0.1875 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (z + 0.5) equal to 0.433012702 and -0.433012702.Subproblem 1
z + 0.5 = 0.433012702 Simplifying z + 0.5 = 0.433012702 Reorder the terms: 0.5 + z = 0.433012702 Solving 0.5 + z = 0.433012702 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 0.433012702 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 0.433012702 + -0.5 z = 0.433012702 + -0.5 Combine like terms: 0.433012702 + -0.5 = -0.066987298 z = -0.066987298 Simplifying z = -0.066987298Subproblem 2
z + 0.5 = -0.433012702 Simplifying z + 0.5 = -0.433012702 Reorder the terms: 0.5 + z = -0.433012702 Solving 0.5 + z = -0.433012702 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -0.433012702 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -0.433012702 + -0.5 z = -0.433012702 + -0.5 Combine like terms: -0.433012702 + -0.5 = -0.933012702 z = -0.933012702 Simplifying z = -0.933012702Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.066987298, -0.933012702}Solution
z = {-0.066987298, -0.933012702}
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