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Simplifying z2 + -81z + 4 = 0 Reorder the terms: 4 + -81z + z2 = 0 Solving 4 + -81z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + -81z + -4 + z2 = 0 + -4 Reorder the terms: 4 + -4 + -81z + z2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -81z + z2 = 0 + -4 -81z + z2 = 0 + -4 Combine like terms: 0 + -4 = -4 -81z + z2 = -4 The z term is -81z. Take half its coefficient (-40.5). Square it (1640.25) and add it to both sides. Add '1640.25' to each side of the equation. -81z + 1640.25 + z2 = -4 + 1640.25 Reorder the terms: 1640.25 + -81z + z2 = -4 + 1640.25 Combine like terms: -4 + 1640.25 = 1636.25 1640.25 + -81z + z2 = 1636.25 Factor a perfect square on the left side: (z + -40.5)(z + -40.5) = 1636.25 Calculate the square root of the right side: 40.45058714 Break this problem into two subproblems by setting (z + -40.5) equal to 40.45058714 and -40.45058714.Subproblem 1
z + -40.5 = 40.45058714 Simplifying z + -40.5 = 40.45058714 Reorder the terms: -40.5 + z = 40.45058714 Solving -40.5 + z = 40.45058714 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '40.5' to each side of the equation. -40.5 + 40.5 + z = 40.45058714 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + z = 40.45058714 + 40.5 z = 40.45058714 + 40.5 Combine like terms: 40.45058714 + 40.5 = 80.95058714 z = 80.95058714 Simplifying z = 80.95058714Subproblem 2
z + -40.5 = -40.45058714 Simplifying z + -40.5 = -40.45058714 Reorder the terms: -40.5 + z = -40.45058714 Solving -40.5 + z = -40.45058714 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '40.5' to each side of the equation. -40.5 + 40.5 + z = -40.45058714 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + z = -40.45058714 + 40.5 z = -40.45058714 + 40.5 Combine like terms: -40.45058714 + 40.5 = 0.04941286 z = 0.04941286 Simplifying z = 0.04941286Solution
The solution to the problem is based on the solutions from the subproblems. z = {80.95058714, 0.04941286}
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