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Simplifying (2x + -1)(5x + -1) + -14x = 0 Reorder the terms: (-1 + 2x)(5x + -1) + -14x = 0 Reorder the terms: (-1 + 2x)(-1 + 5x) + -14x = 0 Multiply (-1 + 2x) * (-1 + 5x) (-1(-1 + 5x) + 2x * (-1 + 5x)) + -14x = 0 ((-1 * -1 + 5x * -1) + 2x * (-1 + 5x)) + -14x = 0 ((1 + -5x) + 2x * (-1 + 5x)) + -14x = 0 (1 + -5x + (-1 * 2x + 5x * 2x)) + -14x = 0 (1 + -5x + (-2x + 10x2)) + -14x = 0 Combine like terms: -5x + -2x = -7x (1 + -7x + 10x2) + -14x = 0 Reorder the terms: 1 + -7x + -14x + 10x2 = 0 Combine like terms: -7x + -14x = -21x 1 + -21x + 10x2 = 0 Solving 1 + -21x + 10x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.1 + -2.1x + x2 = 0.0 Move the constant term to the right: Add '-0.1' to each side of the equation. 0.1 + -2.1x + -0.1 + x2 = 0.0 + -0.1 Reorder the terms: 0.1 + -0.1 + -2.1x + x2 = 0.0 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + -2.1x + x2 = 0.0 + -0.1 -2.1x + x2 = 0.0 + -0.1 Combine like terms: 0.0 + -0.1 = -0.1 -2.1x + x2 = -0.1 The x term is -2.1x. Take half its coefficient (-1.05). Square it (1.1025) and add it to both sides. Add '1.1025' to each side of the equation. -2.1x + 1.1025 + x2 = -0.1 + 1.1025 Reorder the terms: 1.1025 + -2.1x + x2 = -0.1 + 1.1025 Combine like terms: -0.1 + 1.1025 = 1.0025 1.1025 + -2.1x + x2 = 1.0025 Factor a perfect square on the left side: (x + -1.05)(x + -1.05) = 1.0025 Calculate the square root of the right side: 1.00124922 Break this problem into two subproblems by setting (x + -1.05) equal to 1.00124922 and -1.00124922.Subproblem 1
x + -1.05 = 1.00124922 Simplifying x + -1.05 = 1.00124922 Reorder the terms: -1.05 + x = 1.00124922 Solving -1.05 + x = 1.00124922 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + x = 1.00124922 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + x = 1.00124922 + 1.05 x = 1.00124922 + 1.05 Combine like terms: 1.00124922 + 1.05 = 2.05124922 x = 2.05124922 Simplifying x = 2.05124922Subproblem 2
x + -1.05 = -1.00124922 Simplifying x + -1.05 = -1.00124922 Reorder the terms: -1.05 + x = -1.00124922 Solving -1.05 + x = -1.00124922 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + x = -1.00124922 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + x = -1.00124922 + 1.05 x = -1.00124922 + 1.05 Combine like terms: -1.00124922 + 1.05 = 0.04875078 x = 0.04875078 Simplifying x = 0.04875078Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.05124922, 0.04875078}
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