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Simplifying A2 + 2A + 2 + -1 = 167 + -2000A Reorder the terms: 2 + -1 + 2A + A2 = 167 + -2000A Combine like terms: 2 + -1 = 1 1 + 2A + A2 = 167 + -2000A Solving 1 + 2A + A2 = 167 + -2000A Solving for variable 'A'. Reorder the terms: 1 + -167 + 2A + 2000A + A2 = 167 + -2000A + -167 + 2000A Combine like terms: 1 + -167 = -166 -166 + 2A + 2000A + A2 = 167 + -2000A + -167 + 2000A Combine like terms: 2A + 2000A = 2002A -166 + 2002A + A2 = 167 + -2000A + -167 + 2000A Reorder the terms: -166 + 2002A + A2 = 167 + -167 + -2000A + 2000A Combine like terms: 167 + -167 = 0 -166 + 2002A + A2 = 0 + -2000A + 2000A -166 + 2002A + A2 = -2000A + 2000A Combine like terms: -2000A + 2000A = 0 -166 + 2002A + A2 = 0 Begin completing the square. Move the constant term to the right: Add '166' to each side of the equation. -166 + 2002A + 166 + A2 = 0 + 166 Reorder the terms: -166 + 166 + 2002A + A2 = 0 + 166 Combine like terms: -166 + 166 = 0 0 + 2002A + A2 = 0 + 166 2002A + A2 = 0 + 166 Combine like terms: 0 + 166 = 166 2002A + A2 = 166 The A term is 2002A. Take half its coefficient (1001). Square it (1002001) and add it to both sides. Add '1002001' to each side of the equation. 2002A + 1002001 + A2 = 166 + 1002001 Reorder the terms: 1002001 + 2002A + A2 = 166 + 1002001 Combine like terms: 166 + 1002001 = 1002167 1002001 + 2002A + A2 = 1002167 Factor a perfect square on the left side: (A + 1001)(A + 1001) = 1002167 Calculate the square root of the right side: 1001.082913649 Break this problem into two subproblems by setting (A + 1001) equal to 1001.082913649 and -1001.082913649.Subproblem 1
A + 1001 = 1001.082913649 Simplifying A + 1001 = 1001.082913649 Reorder the terms: 1001 + A = 1001.082913649 Solving 1001 + A = 1001.082913649 Solving for variable 'A'. Move all terms containing A to the left, all other terms to the right. Add '-1001' to each side of the equation. 1001 + -1001 + A = 1001.082913649 + -1001 Combine like terms: 1001 + -1001 = 0 0 + A = 1001.082913649 + -1001 A = 1001.082913649 + -1001 Combine like terms: 1001.082913649 + -1001 = 0.082913649 A = 0.082913649 Simplifying A = 0.082913649Subproblem 2
A + 1001 = -1001.082913649 Simplifying A + 1001 = -1001.082913649 Reorder the terms: 1001 + A = -1001.082913649 Solving 1001 + A = -1001.082913649 Solving for variable 'A'. Move all terms containing A to the left, all other terms to the right. Add '-1001' to each side of the equation. 1001 + -1001 + A = -1001.082913649 + -1001 Combine like terms: 1001 + -1001 = 0 0 + A = -1001.082913649 + -1001 A = -1001.082913649 + -1001 Combine like terms: -1001.082913649 + -1001 = -2002.082913649 A = -2002.082913649 Simplifying A = -2002.082913649Solution
The solution to the problem is based on the solutions from the subproblems. A = {0.082913649, -2002.082913649}
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