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Simplifying z2 + -10z = -17 Reorder the terms: -10z + z2 = -17 Solving -10z + z2 = -17 Solving for variable 'z'. Reorder the terms: 17 + -10z + z2 = -17 + 17 Combine like terms: -17 + 17 = 0 17 + -10z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-17' to each side of the equation. 17 + -10z + -17 + z2 = 0 + -17 Reorder the terms: 17 + -17 + -10z + z2 = 0 + -17 Combine like terms: 17 + -17 = 0 0 + -10z + z2 = 0 + -17 -10z + z2 = 0 + -17 Combine like terms: 0 + -17 = -17 -10z + z2 = -17 The z term is -10z. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10z + 25 + z2 = -17 + 25 Reorder the terms: 25 + -10z + z2 = -17 + 25 Combine like terms: -17 + 25 = 8 25 + -10z + z2 = 8 Factor a perfect square on the left side: (z + -5)(z + -5) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (z + -5) equal to 2.828427125 and -2.828427125.Subproblem 1
z + -5 = 2.828427125 Simplifying z + -5 = 2.828427125 Reorder the terms: -5 + z = 2.828427125 Solving -5 + z = 2.828427125 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 + z = 2.828427125 + 5 Combine like terms: -5 + 5 = 0 0 + z = 2.828427125 + 5 z = 2.828427125 + 5 Combine like terms: 2.828427125 + 5 = 7.828427125 z = 7.828427125 Simplifying z = 7.828427125Subproblem 2
z + -5 = -2.828427125 Simplifying z + -5 = -2.828427125 Reorder the terms: -5 + z = -2.828427125 Solving -5 + z = -2.828427125 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 + z = -2.828427125 + 5 Combine like terms: -5 + 5 = 0 0 + z = -2.828427125 + 5 z = -2.828427125 + 5 Combine like terms: -2.828427125 + 5 = 2.171572875 z = 2.171572875 Simplifying z = 2.171572875Solution
The solution to the problem is based on the solutions from the subproblems. z = {7.828427125, 2.171572875}
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