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Simplifying z2 + -6z + 7z + -48 = 0 Reorder the terms: -48 + -6z + 7z + z2 = 0 Combine like terms: -6z + 7z = 1z -48 + 1z + z2 = 0 Solving -48 + 1z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '48' to each side of the equation. -48 + 1z + 48 + z2 = 0 + 48 Reorder the terms: -48 + 48 + 1z + z2 = 0 + 48 Combine like terms: -48 + 48 = 0 0 + 1z + z2 = 0 + 48 1z + z2 = 0 + 48 Combine like terms: 0 + 48 = 48 1z + z2 = 48 The z term is 1z. 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. z + 0.25 + z2 = 48 + 0.25 Reorder the terms: 0.25 + z + z2 = 48 + 0.25 Combine like terms: 48 + 0.25 = 48.25 0.25 + z + z2 = 48.25 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 48.25 Calculate the square root of the right side: 6.946221995 Break this problem into two subproblems by setting (z + 0.5) equal to 6.946221995 and -6.946221995.Subproblem 1
z + 0.5 = 6.946221995 Simplifying z + 0.5 = 6.946221995 Reorder the terms: 0.5 + z = 6.946221995 Solving 0.5 + z = 6.946221995 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 = 6.946221995 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 6.946221995 + -0.5 z = 6.946221995 + -0.5 Combine like terms: 6.946221995 + -0.5 = 6.446221995 z = 6.446221995 Simplifying z = 6.446221995Subproblem 2
z + 0.5 = -6.946221995 Simplifying z + 0.5 = -6.946221995 Reorder the terms: 0.5 + z = -6.946221995 Solving 0.5 + z = -6.946221995 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 = -6.946221995 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -6.946221995 + -0.5 z = -6.946221995 + -0.5 Combine like terms: -6.946221995 + -0.5 = -7.446221995 z = -7.446221995 Simplifying z = -7.446221995Solution
The solution to the problem is based on the solutions from the subproblems. z = {6.446221995, -7.446221995}
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