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Simplifying z4 + -10z2 + 1 = 0 Reorder the terms: 1 + -10z2 + z4 = 0 Solving 1 + -10z2 + z4 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -10z2 + -1 + z4 = 0 + -1 Reorder the terms: 1 + -1 + -10z2 + z4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -10z2 + z4 = 0 + -1 -10z2 + z4 = 0 + -1 Combine like terms: 0 + -1 = -1 -10z2 + z4 = -1 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 = -1 + 25 Reorder the terms: 25 + -10z2 + z4 = -1 + 25 Combine like terms: -1 + 25 = 24 25 + -10z2 + z4 = 24 Factor a perfect square on the left side: (z2 + -5)(z2 + -5) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (z2 + -5) equal to 4.898979486 and -4.898979486.Subproblem 1
z2 + -5 = 4.898979486 Simplifying z2 + -5 = 4.898979486 Reorder the terms: -5 + z2 = 4.898979486 Solving -5 + z2 = 4.898979486 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 = 4.898979486 + 5 Combine like terms: -5 + 5 = 0 0 + z2 = 4.898979486 + 5 z2 = 4.898979486 + 5 Combine like terms: 4.898979486 + 5 = 9.898979486 z2 = 9.898979486 Simplifying z2 = 9.898979486 Take the square root of each side: z = {-3.14626437, 3.14626437}Subproblem 2
z2 + -5 = -4.898979486 Simplifying z2 + -5 = -4.898979486 Reorder the terms: -5 + z2 = -4.898979486 Solving -5 + z2 = -4.898979486 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 = -4.898979486 + 5 Combine like terms: -5 + 5 = 0 0 + z2 = -4.898979486 + 5 z2 = -4.898979486 + 5 Combine like terms: -4.898979486 + 5 = 0.101020514 z2 = 0.101020514 Simplifying z2 = 0.101020514 Take the square root of each side: z = {-0.317837245, 0.317837245}Solution
The solution to the problem is based on the solutions from the subproblems. z = {-3.14626437, 3.14626437, -0.317837245, 0.317837245}
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