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Simplifying z2 + -18z = 31 Reorder the terms: -18z + z2 = 31 Solving -18z + z2 = 31 Solving for variable 'z'. Reorder the terms: -31 + -18z + z2 = 31 + -31 Combine like terms: 31 + -31 = 0 -31 + -18z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '31' to each side of the equation. -31 + -18z + 31 + z2 = 0 + 31 Reorder the terms: -31 + 31 + -18z + z2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + -18z + z2 = 0 + 31 -18z + z2 = 0 + 31 Combine like terms: 0 + 31 = 31 -18z + z2 = 31 The z term is -18z. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18z + 81 + z2 = 31 + 81 Reorder the terms: 81 + -18z + z2 = 31 + 81 Combine like terms: 31 + 81 = 112 81 + -18z + z2 = 112 Factor a perfect square on the left side: (z + -9)(z + -9) = 112 Calculate the square root of the right side: 10.583005244 Break this problem into two subproblems by setting (z + -9) equal to 10.583005244 and -10.583005244.Subproblem 1
z + -9 = 10.583005244 Simplifying z + -9 = 10.583005244 Reorder the terms: -9 + z = 10.583005244 Solving -9 + z = 10.583005244 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + z = 10.583005244 + 9 Combine like terms: -9 + 9 = 0 0 + z = 10.583005244 + 9 z = 10.583005244 + 9 Combine like terms: 10.583005244 + 9 = 19.583005244 z = 19.583005244 Simplifying z = 19.583005244Subproblem 2
z + -9 = -10.583005244 Simplifying z + -9 = -10.583005244 Reorder the terms: -9 + z = -10.583005244 Solving -9 + z = -10.583005244 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + z = -10.583005244 + 9 Combine like terms: -9 + 9 = 0 0 + z = -10.583005244 + 9 z = -10.583005244 + 9 Combine like terms: -10.583005244 + 9 = -1.583005244 z = -1.583005244 Simplifying z = -1.583005244Solution
The solution to the problem is based on the solutions from the subproblems. z = {19.583005244, -1.583005244}
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