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Simplifying 2Z2 = 7Z + 6 Reorder the terms: 2Z2 = 6 + 7Z Solving 2Z2 = 6 + 7Z Solving for variable 'Z'. Reorder the terms: -6 + -7Z + 2Z2 = 6 + 7Z + -6 + -7Z Reorder the terms: -6 + -7Z + 2Z2 = 6 + -6 + 7Z + -7Z Combine like terms: 6 + -6 = 0 -6 + -7Z + 2Z2 = 0 + 7Z + -7Z -6 + -7Z + 2Z2 = 7Z + -7Z Combine like terms: 7Z + -7Z = 0 -6 + -7Z + 2Z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3 + -3.5Z + Z2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -3.5Z + 3 + Z2 = 0 + 3 Reorder the terms: -3 + 3 + -3.5Z + Z2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -3.5Z + Z2 = 0 + 3 -3.5Z + Z2 = 0 + 3 Combine like terms: 0 + 3 = 3 -3.5Z + Z2 = 3 The Z term is -3.5Z. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5Z + 3.0625 + Z2 = 3 + 3.0625 Reorder the terms: 3.0625 + -3.5Z + Z2 = 3 + 3.0625 Combine like terms: 3 + 3.0625 = 6.0625 3.0625 + -3.5Z + Z2 = 6.0625 Factor a perfect square on the left side: (Z + -1.75)(Z + -1.75) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (Z + -1.75) equal to 2.46221445 and -2.46221445.Subproblem 1
Z + -1.75 = 2.46221445 Simplifying Z + -1.75 = 2.46221445 Reorder the terms: -1.75 + Z = 2.46221445 Solving -1.75 + Z = 2.46221445 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + Z = 2.46221445 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + Z = 2.46221445 + 1.75 Z = 2.46221445 + 1.75 Combine like terms: 2.46221445 + 1.75 = 4.21221445 Z = 4.21221445 Simplifying Z = 4.21221445Subproblem 2
Z + -1.75 = -2.46221445 Simplifying Z + -1.75 = -2.46221445 Reorder the terms: -1.75 + Z = -2.46221445 Solving -1.75 + Z = -2.46221445 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + Z = -2.46221445 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + Z = -2.46221445 + 1.75 Z = -2.46221445 + 1.75 Combine like terms: -2.46221445 + 1.75 = -0.71221445 Z = -0.71221445 Simplifying Z = -0.71221445Solution
The solution to the problem is based on the solutions from the subproblems. Z = {4.21221445, -0.71221445}
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