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Simplifying 30 = (1 + 0.0666666x) * x Reorder the terms for easier multiplication: 30 = x(1 + 0.0666666x) 30 = (1 * x + 0.0666666x * x) 30 = (1x + 0.0666666x2) Solving 30 = 1x + 0.0666666x2 Solving for variable 'x'. Reorder the terms: 30 + -1x + -0.0666666x2 = 1x + -1x + 0.0666666x2 + -0.0666666x2 Combine like terms: 1x + -1x = 0 30 + -1x + -0.0666666x2 = 0 + 0.0666666x2 + -0.0666666x2 30 + -1x + -0.0666666x2 = 0.0666666x2 + -0.0666666x2 Combine like terms: 0.0666666x2 + -0.0666666x2 = 0.0000000 30 + -1x + -0.0666666x2 = 0.0000000 Begin completing the square. Divide all terms by -0.0666666 the coefficient of the squared term: Divide each side by '-0.0666666'. -450.00045 + 15.000015x + x2 = 0 Move the constant term to the right: Add '450.00045' to each side of the equation. -450.00045 + 15.000015x + 450.00045 + x2 = 0 + 450.00045 Reorder the terms: -450.00045 + 450.00045 + 15.000015x + x2 = 0 + 450.00045 Combine like terms: -450.00045 + 450.00045 = 0.00000 0.00000 + 15.000015x + x2 = 0 + 450.00045 15.000015x + x2 = 0 + 450.00045 Combine like terms: 0 + 450.00045 = 450.00045 15.000015x + x2 = 450.00045 The x term is 15.000015x. Take half its coefficient (7.5000075). Square it (56.25011250) and add it to both sides. Add '56.25011250' to each side of the equation. 15.000015x + 56.25011250 + x2 = 450.00045 + 56.25011250 Reorder the terms: 56.25011250 + 15.000015x + x2 = 450.00045 + 56.25011250 Combine like terms: 450.00045 + 56.25011250 = 506.2505625 56.25011250 + 15.000015x + x2 = 506.2505625 Factor a perfect square on the left side: (x + 7.5000075)(x + 7.5000075) = 506.2505625 Calculate the square root of the right side: 22.5000125 Break this problem into two subproblems by setting (x + 7.5000075) equal to 22.5000125 and -22.5000125.Subproblem 1
x + 7.5000075 = 22.5000125 Simplifying x + 7.5000075 = 22.5000125 Reorder the terms: 7.5000075 + x = 22.5000125 Solving 7.5000075 + x = 22.5000125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5000075' to each side of the equation. 7.5000075 + -7.5000075 + x = 22.5000125 + -7.5000075 Combine like terms: 7.5000075 + -7.5000075 = 0.0000000 0.0000000 + x = 22.5000125 + -7.5000075 x = 22.5000125 + -7.5000075 Combine like terms: 22.5000125 + -7.5000075 = 15.000005 x = 15.000005 Simplifying x = 15.000005Subproblem 2
x + 7.5000075 = -22.5000125 Simplifying x + 7.5000075 = -22.5000125 Reorder the terms: 7.5000075 + x = -22.5000125 Solving 7.5000075 + x = -22.5000125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5000075' to each side of the equation. 7.5000075 + -7.5000075 + x = -22.5000125 + -7.5000075 Combine like terms: 7.5000075 + -7.5000075 = 0.0000000 0.0000000 + x = -22.5000125 + -7.5000075 x = -22.5000125 + -7.5000075 Combine like terms: -22.5000125 + -7.5000075 = -30.00002 x = -30.00002 Simplifying x = -30.00002Solution
The solution to the problem is based on the solutions from the subproblems. x = {15.000005, -30.00002}
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