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Simplifying z4 + -8z2 + 11 = 0 Reorder the terms: 11 + -8z2 + z4 = 0 Solving 11 + -8z2 + z4 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-11' to each side of the equation. 11 + -8z2 + -11 + z4 = 0 + -11 Reorder the terms: 11 + -11 + -8z2 + z4 = 0 + -11 Combine like terms: 11 + -11 = 0 0 + -8z2 + z4 = 0 + -11 -8z2 + z4 = 0 + -11 Combine like terms: 0 + -11 = -11 -8z2 + z4 = -11 The z term is -8z2. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8z2 + 16 + z4 = -11 + 16 Reorder the terms: 16 + -8z2 + z4 = -11 + 16 Combine like terms: -11 + 16 = 5 16 + -8z2 + z4 = 5 Factor a perfect square on the left side: (z2 + -4)(z2 + -4) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (z2 + -4) equal to 2.236067978 and -2.236067978.Subproblem 1
z2 + -4 = 2.236067978 Simplifying z2 + -4 = 2.236067978 Reorder the terms: -4 + z2 = 2.236067978 Solving -4 + z2 = 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 + z2 = 2.236067978 + 4 Combine like terms: -4 + 4 = 0 0 + z2 = 2.236067978 + 4 z2 = 2.236067978 + 4 Combine like terms: 2.236067978 + 4 = 6.236067978 z2 = 6.236067978 Simplifying z2 = 6.236067978 Take the square root of each side: z = {-2.497212041, 2.497212041}Subproblem 2
z2 + -4 = -2.236067978 Simplifying z2 + -4 = -2.236067978 Reorder the terms: -4 + z2 = -2.236067978 Solving -4 + z2 = -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 + z2 = -2.236067978 + 4 Combine like terms: -4 + 4 = 0 0 + z2 = -2.236067978 + 4 z2 = -2.236067978 + 4 Combine like terms: -2.236067978 + 4 = 1.763932022 z2 = 1.763932022 Simplifying z2 = 1.763932022 Take the square root of each side: z = {-1.328131026, 1.328131026}Solution
The solution to the problem is based on the solutions from the subproblems. z = {-2.497212041, 2.497212041, -1.328131026, 1.328131026}
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