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Simplifying 0.65(1 + -1x)(2 + -1x) = x Multiply (1 + -1x) * (2 + -1x) 0.65(1(2 + -1x) + -1x * (2 + -1x)) = x 0.65((2 * 1 + -1x * 1) + -1x * (2 + -1x)) = x 0.65((2 + -1x) + -1x * (2 + -1x)) = x 0.65(2 + -1x + (2 * -1x + -1x * -1x)) = x 0.65(2 + -1x + (-2x + 1x2)) = x Combine like terms: -1x + -2x = -3x 0.65(2 + -3x + 1x2) = x (2 * 0.65 + -3x * 0.65 + 1x2 * 0.65) = x (1.3 + -1.95x + 0.65x2) = x Solving 1.3 + -1.95x + 0.65x2 = x Solving for variable 'x'. Reorder the terms: 1.3 + -1.95x + -1x + 0.65x2 = x + -1x Combine like terms: -1.95x + -1x = -2.95x 1.3 + -2.95x + 0.65x2 = x + -1x Combine like terms: x + -1x = 0 1.3 + -2.95x + 0.65x2 = 0 Begin completing the square. Divide all terms by 0.65 the coefficient of the squared term: Divide each side by '0.65'. 2 + -4.538461538x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4.538461538x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -4.538461538x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4.538461538x + x2 = 0 + -2 -4.538461538x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -4.538461538x + x2 = -2 The x term is -4.538461538x. Take half its coefficient (-2.269230769). Square it (5.149408283) and add it to both sides. Add '5.149408283' to each side of the equation. -4.538461538x + 5.149408283 + x2 = -2 + 5.149408283 Reorder the terms: 5.149408283 + -4.538461538x + x2 = -2 + 5.149408283 Combine like terms: -2 + 5.149408283 = 3.149408283 5.149408283 + -4.538461538x + x2 = 3.149408283 Factor a perfect square on the left side: (x + -2.269230769)(x + -2.269230769) = 3.149408283 Calculate the square root of the right side: 1.77465723 Break this problem into two subproblems by setting (x + -2.269230769) equal to 1.77465723 and -1.77465723.Subproblem 1
x + -2.269230769 = 1.77465723 Simplifying x + -2.269230769 = 1.77465723 Reorder the terms: -2.269230769 + x = 1.77465723 Solving -2.269230769 + x = 1.77465723 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.269230769' to each side of the equation. -2.269230769 + 2.269230769 + x = 1.77465723 + 2.269230769 Combine like terms: -2.269230769 + 2.269230769 = 0.000000000 0.000000000 + x = 1.77465723 + 2.269230769 x = 1.77465723 + 2.269230769 Combine like terms: 1.77465723 + 2.269230769 = 4.043887999 x = 4.043887999 Simplifying x = 4.043887999Subproblem 2
x + -2.269230769 = -1.77465723 Simplifying x + -2.269230769 = -1.77465723 Reorder the terms: -2.269230769 + x = -1.77465723 Solving -2.269230769 + x = -1.77465723 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.269230769' to each side of the equation. -2.269230769 + 2.269230769 + x = -1.77465723 + 2.269230769 Combine like terms: -2.269230769 + 2.269230769 = 0.000000000 0.000000000 + x = -1.77465723 + 2.269230769 x = -1.77465723 + 2.269230769 Combine like terms: -1.77465723 + 2.269230769 = 0.494573539 x = 0.494573539 Simplifying x = 0.494573539Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.043887999, 0.494573539}
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