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Simplifying z2 = -5z + -1 Reorder the terms: z2 = -1 + -5z Solving z2 = -1 + -5z Solving for variable 'z'. Reorder the terms: 1 + 5z + z2 = -1 + -5z + 1 + 5z Reorder the terms: 1 + 5z + z2 = -1 + 1 + -5z + 5z Combine like terms: -1 + 1 = 0 1 + 5z + z2 = 0 + -5z + 5z 1 + 5z + z2 = -5z + 5z Combine like terms: -5z + 5z = 0 1 + 5z + z2 = 0 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 = -0.208712153 z = -0.208712153 Simplifying z = -0.208712153Subproblem 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 = -4.791287847 z = -4.791287847 Simplifying z = -4.791287847Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.208712153, -4.791287847}
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