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Simplifying 82[0.083 + -1x][0.083 + -1x] = 4x2 Multiply [0.083 + -1x] * [0.083 + -1x] 82[0.083[0.083 + -1x] + -1x * [0.083 + -1x]] = 4x2 82[[0.083 * 0.083 + -1x * 0.083] + -1x * [0.083 + -1x]] = 4x2 82[[0.006889 + -0.083x] + -1x * [0.083 + -1x]] = 4x2 82[0.006889 + -0.083x + [0.083 * -1x + -1x * -1x]] = 4x2 82[0.006889 + -0.083x + [-0.083x + 1x2]] = 4x2 Combine like terms: -0.083x + -0.083x = -0.166x 82[0.006889 + -0.166x + 1x2] = 4x2 [0.006889 * 82 + -0.166x * 82 + 1x2 * 82] = 4x2 [0.564898 + -13.612x + 82x2] = 4x2 Solving 0.564898 + -13.612x + 82x2 = 4x2 Solving for variable 'x'. Combine like terms: 82x2 + -4x2 = 78x2 0.564898 + -13.612x + 78x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 0.564898 + -13.612x + 78x2 = 0 Begin completing the square. Divide all terms by 78 the coefficient of the squared term: Divide each side by '78'. 0.007242282051 + -0.1745128205x + x2 = 0 Move the constant term to the right: Add '-0.007242282051' to each side of the equation. 0.007242282051 + -0.1745128205x + -0.007242282051 + x2 = 0 + -0.007242282051 Reorder the terms: 0.007242282051 + -0.007242282051 + -0.1745128205x + x2 = 0 + -0.007242282051 Combine like terms: 0.007242282051 + -0.007242282051 = 0.000000000000 0.000000000000 + -0.1745128205x + x2 = 0 + -0.007242282051 -0.1745128205x + x2 = 0 + -0.007242282051 Combine like terms: 0 + -0.007242282051 = -0.007242282051 -0.1745128205x + x2 = -0.007242282051 The x term is -0.1745128205x. Take half its coefficient (-0.08725641025). Square it (0.007613681130) and add it to both sides. Add '0.007613681130' to each side of the equation. -0.1745128205x + 0.007613681130 + x2 = -0.007242282051 + 0.007613681130 Reorder the terms: 0.007613681130 + -0.1745128205x + x2 = -0.007242282051 + 0.007613681130 Combine like terms: -0.007242282051 + 0.007613681130 = 0.000371399079 0.007613681130 + -0.1745128205x + x2 = 0.000371399079 Factor a perfect square on the left side: (x + -0.08725641025)(x + -0.08725641025) = 0.000371399079 Calculate the square root of the right side: 0.019271717 Break this problem into two subproblems by setting (x + -0.08725641025) equal to 0.019271717 and -0.019271717.Subproblem 1
x + -0.08725641025 = 0.019271717 Simplifying x + -0.08725641025 = 0.019271717 Reorder the terms: -0.08725641025 + x = 0.019271717 Solving -0.08725641025 + x = 0.019271717 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.08725641025' to each side of the equation. -0.08725641025 + 0.08725641025 + x = 0.019271717 + 0.08725641025 Combine like terms: -0.08725641025 + 0.08725641025 = 0.00000000000 0.00000000000 + x = 0.019271717 + 0.08725641025 x = 0.019271717 + 0.08725641025 Combine like terms: 0.019271717 + 0.08725641025 = 0.10652812725 x = 0.10652812725 Simplifying x = 0.10652812725Subproblem 2
x + -0.08725641025 = -0.019271717 Simplifying x + -0.08725641025 = -0.019271717 Reorder the terms: -0.08725641025 + x = -0.019271717 Solving -0.08725641025 + x = -0.019271717 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.08725641025' to each side of the equation. -0.08725641025 + 0.08725641025 + x = -0.019271717 + 0.08725641025 Combine like terms: -0.08725641025 + 0.08725641025 = 0.00000000000 0.00000000000 + x = -0.019271717 + 0.08725641025 x = -0.019271717 + 0.08725641025 Combine like terms: -0.019271717 + 0.08725641025 = 0.06798469325 x = 0.06798469325 Simplifying x = 0.06798469325Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.10652812725, 0.06798469325}
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