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Simplifying z4 + -10z2 + 5 = 0 Reorder the terms: 5 + -10z2 + z4 = 0 Solving 5 + -10z2 + z4 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + -10z2 + -5 + z4 = 0 + -5 Reorder the terms: 5 + -5 + -10z2 + z4 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -10z2 + z4 = 0 + -5 -10z2 + z4 = 0 + -5 Combine like terms: 0 + -5 = -5 -10z2 + z4 = -5 The z term is -10z2. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10z2 + 25 + z4 = -5 + 25 Reorder the terms: 25 + -10z2 + z4 = -5 + 25 Combine like terms: -5 + 25 = 20 25 + -10z2 + z4 = 20 Factor a perfect square on the left side: (z2 + -5)(z2 + -5) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (z2 + -5) equal to 4.472135955 and -4.472135955.Subproblem 1
z2 + -5 = 4.472135955 Simplifying z2 + -5 = 4.472135955 Reorder the terms: -5 + z2 = 4.472135955 Solving -5 + z2 = 4.472135955 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z2 = 4.472135955 + 5 Combine like terms: -5 + 5 = 0 0 + z2 = 4.472135955 + 5 z2 = 4.472135955 + 5 Combine like terms: 4.472135955 + 5 = 9.472135955 z2 = 9.472135955 Simplifying z2 = 9.472135955 Take the square root of each side: z = {-3.077683537, 3.077683537}Subproblem 2
z2 + -5 = -4.472135955 Simplifying z2 + -5 = -4.472135955 Reorder the terms: -5 + z2 = -4.472135955 Solving -5 + z2 = -4.472135955 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z2 = -4.472135955 + 5 Combine like terms: -5 + 5 = 0 0 + z2 = -4.472135955 + 5 z2 = -4.472135955 + 5 Combine like terms: -4.472135955 + 5 = 0.527864045 z2 = 0.527864045 Simplifying z2 = 0.527864045 Take the square root of each side: z = {-0.726542528, 0.726542528}Solution
The solution to the problem is based on the solutions from the subproblems. z = {-3.077683537, 3.077683537, -0.726542528, 0.726542528}
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