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Simplifying z2 + -5z + 1 = 0 Reorder the terms: 1 + -5z + z2 = 0 Solving 1 + -5z + z2 = 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 + -5z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + -5z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -5z + z2 = 0 + -1 -5z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 -5z + z2 = -1 The z term is -5z. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5z + 6.25 + z2 = -1 + 6.25 Reorder the terms: 6.25 + -5z + z2 = -1 + 6.25 Combine like terms: -1 + 6.25 = 5.25 6.25 + -5z + z2 = 5.25 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (z + -2.5) equal to 2.291287847 and -2.291287847.Subproblem 1
z + -2.5 = 2.291287847 Simplifying z + -2.5 = 2.291287847 Reorder the terms: -2.5 + z = 2.291287847 Solving -2.5 + z = 2.291287847 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 2.291287847 + 2.5 z = 2.291287847 + 2.5 Combine like terms: 2.291287847 + 2.5 = 4.791287847 z = 4.791287847 Simplifying z = 4.791287847Subproblem 2
z + -2.5 = -2.291287847 Simplifying z + -2.5 = -2.291287847 Reorder the terms: -2.5 + z = -2.291287847 Solving -2.5 + z = -2.291287847 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -2.291287847 + 2.5 z = -2.291287847 + 2.5 Combine like terms: -2.291287847 + 2.5 = 0.208712153 z = 0.208712153 Simplifying z = 0.208712153Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.791287847, 0.208712153}
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