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Simplifying z2 + -8z = -11 Reorder the terms: -8z + z2 = -11 Solving -8z + z2 = -11 Solving for variable 'z'. Reorder the terms: 11 + -8z + z2 = -11 + 11 Combine like terms: -11 + 11 = 0 11 + -8z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-11' to each side of the equation. 11 + -8z + -11 + z2 = 0 + -11 Reorder the terms: 11 + -11 + -8z + z2 = 0 + -11 Combine like terms: 11 + -11 = 0 0 + -8z + z2 = 0 + -11 -8z + z2 = 0 + -11 Combine like terms: 0 + -11 = -11 -8z + z2 = -11 The z term is -8z. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8z + 16 + z2 = -11 + 16 Reorder the terms: 16 + -8z + z2 = -11 + 16 Combine like terms: -11 + 16 = 5 16 + -8z + z2 = 5 Factor a perfect square on the left side: (z + -4)(z + -4) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (z + -4) equal to 2.236067978 and -2.236067978.Subproblem 1
z + -4 = 2.236067978 Simplifying z + -4 = 2.236067978 Reorder the terms: -4 + z = 2.236067978 Solving -4 + z = 2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 2.236067978 + 4 Combine like terms: -4 + 4 = 0 0 + z = 2.236067978 + 4 z = 2.236067978 + 4 Combine like terms: 2.236067978 + 4 = 6.236067978 z = 6.236067978 Simplifying z = 6.236067978Subproblem 2
z + -4 = -2.236067978 Simplifying z + -4 = -2.236067978 Reorder the terms: -4 + z = -2.236067978 Solving -4 + z = -2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -2.236067978 + 4 Combine like terms: -4 + 4 = 0 0 + z = -2.236067978 + 4 z = -2.236067978 + 4 Combine like terms: -2.236067978 + 4 = 1.763932022 z = 1.763932022 Simplifying z = 1.763932022Solution
The solution to the problem is based on the solutions from the subproblems. z = {6.236067978, 1.763932022}
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