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Simplifying z2 + -81z + 196 = 0 Reorder the terms: 196 + -81z + z2 = 0 Solving 196 + -81z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-196' to each side of the equation. 196 + -81z + -196 + z2 = 0 + -196 Reorder the terms: 196 + -196 + -81z + z2 = 0 + -196 Combine like terms: 196 + -196 = 0 0 + -81z + z2 = 0 + -196 -81z + z2 = 0 + -196 Combine like terms: 0 + -196 = -196 -81z + z2 = -196 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 = -196 + 1640.25 Reorder the terms: 1640.25 + -81z + z2 = -196 + 1640.25 Combine like terms: -196 + 1640.25 = 1444.25 1640.25 + -81z + z2 = 1444.25 Factor a perfect square on the left side: (z + -40.5)(z + -40.5) = 1444.25 Calculate the square root of the right side: 38.003289331 Break this problem into two subproblems by setting (z + -40.5) equal to 38.003289331 and -38.003289331.Subproblem 1
z + -40.5 = 38.003289331 Simplifying z + -40.5 = 38.003289331 Reorder the terms: -40.5 + z = 38.003289331 Solving -40.5 + z = 38.003289331 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 = 38.003289331 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + z = 38.003289331 + 40.5 z = 38.003289331 + 40.5 Combine like terms: 38.003289331 + 40.5 = 78.503289331 z = 78.503289331 Simplifying z = 78.503289331Subproblem 2
z + -40.5 = -38.003289331 Simplifying z + -40.5 = -38.003289331 Reorder the terms: -40.5 + z = -38.003289331 Solving -40.5 + z = -38.003289331 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 = -38.003289331 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + z = -38.003289331 + 40.5 z = -38.003289331 + 40.5 Combine like terms: -38.003289331 + 40.5 = 2.496710669 z = 2.496710669 Simplifying z = 2.496710669Solution
The solution to the problem is based on the solutions from the subproblems. z = {78.503289331, 2.496710669}
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