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Simplifying (2)(0.5Z * 0.5 + 7) = (Z)(Z + -5) Reorder the terms for easier multiplication: 2(0.5 * 0.5Z + 7) = (Z)(Z + -5) Multiply 0.5 * 0.5 2(0.25Z + 7) = (Z)(Z + -5) Reorder the terms: 2(7 + 0.25Z) = (Z)(Z + -5) (7 * 2 + 0.25Z * 2) = (Z)(Z + -5) (14 + 0.5Z) = (Z)(Z + -5) Reorder the terms: 14 + 0.5Z = Z(-5 + Z) 14 + 0.5Z = (-5 * Z + Z * Z) 14 + 0.5Z = (-5Z + Z2) Solving 14 + 0.5Z = -5Z + Z2 Solving for variable 'Z'. Combine like terms: 0.5Z + 5Z = 5.5Z 14 + 5.5Z + -1Z2 = -5Z + Z2 + 5Z + -1Z2 Reorder the terms: 14 + 5.5Z + -1Z2 = -5Z + 5Z + Z2 + -1Z2 Combine like terms: -5Z + 5Z = 0 14 + 5.5Z + -1Z2 = 0 + Z2 + -1Z2 14 + 5.5Z + -1Z2 = Z2 + -1Z2 Combine like terms: Z2 + -1Z2 = 0 14 + 5.5Z + -1Z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -14 + -5.5Z + Z2 = 0 Move the constant term to the right: Add '14' to each side of the equation. -14 + -5.5Z + 14 + Z2 = 0 + 14 Reorder the terms: -14 + 14 + -5.5Z + Z2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -5.5Z + Z2 = 0 + 14 -5.5Z + Z2 = 0 + 14 Combine like terms: 0 + 14 = 14 -5.5Z + Z2 = 14 The Z term is -5.5Z. Take half its coefficient (-2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. -5.5Z + 7.5625 + Z2 = 14 + 7.5625 Reorder the terms: 7.5625 + -5.5Z + Z2 = 14 + 7.5625 Combine like terms: 14 + 7.5625 = 21.5625 7.5625 + -5.5Z + Z2 = 21.5625 Factor a perfect square on the left side: (Z + -2.75)(Z + -2.75) = 21.5625 Calculate the square root of the right side: 4.643543905 Break this problem into two subproblems by setting (Z + -2.75) equal to 4.643543905 and -4.643543905.Subproblem 1
Z + -2.75 = 4.643543905 Simplifying Z + -2.75 = 4.643543905 Reorder the terms: -2.75 + Z = 4.643543905 Solving -2.75 + Z = 4.643543905 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + Z = 4.643543905 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + Z = 4.643543905 + 2.75 Z = 4.643543905 + 2.75 Combine like terms: 4.643543905 + 2.75 = 7.393543905 Z = 7.393543905 Simplifying Z = 7.393543905Subproblem 2
Z + -2.75 = -4.643543905 Simplifying Z + -2.75 = -4.643543905 Reorder the terms: -2.75 + Z = -4.643543905 Solving -2.75 + Z = -4.643543905 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + Z = -4.643543905 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + Z = -4.643543905 + 2.75 Z = -4.643543905 + 2.75 Combine like terms: -4.643543905 + 2.75 = -1.893543905 Z = -1.893543905 Simplifying Z = -1.893543905Solution
The solution to the problem is based on the solutions from the subproblems. Z = {7.393543905, -1.893543905}
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