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Simplifying 30 = x(0.09x + 1) Reorder the terms: 30 = x(1 + 0.09x) 30 = (1 * x + 0.09x * x) 30 = (1x + 0.09x2) Solving 30 = 1x + 0.09x2 Solving for variable 'x'. Reorder the terms: 30 + -1x + -0.09x2 = 1x + -1x + 0.09x2 + -0.09x2 Combine like terms: 1x + -1x = 0 30 + -1x + -0.09x2 = 0 + 0.09x2 + -0.09x2 30 + -1x + -0.09x2 = 0.09x2 + -0.09x2 Combine like terms: 0.09x2 + -0.09x2 = 0.00 30 + -1x + -0.09x2 = 0.00 Begin completing the square. Divide all terms by -0.09 the coefficient of the squared term: Divide each side by '-0.09'. -333.3333333 + 11.11111111x + x2 = 0 Move the constant term to the right: Add '333.3333333' to each side of the equation. -333.3333333 + 11.11111111x + 333.3333333 + x2 = 0 + 333.3333333 Reorder the terms: -333.3333333 + 333.3333333 + 11.11111111x + x2 = 0 + 333.3333333 Combine like terms: -333.3333333 + 333.3333333 = 0.0000000 0.0000000 + 11.11111111x + x2 = 0 + 333.3333333 11.11111111x + x2 = 0 + 333.3333333 Combine like terms: 0 + 333.3333333 = 333.3333333 11.11111111x + x2 = 333.3333333 The x term is 11.11111111x. Take half its coefficient (5.555555555). Square it (30.86419752) and add it to both sides. Add '30.86419752' to each side of the equation. 11.11111111x + 30.86419752 + x2 = 333.3333333 + 30.86419752 Reorder the terms: 30.86419752 + 11.11111111x + x2 = 333.3333333 + 30.86419752 Combine like terms: 333.3333333 + 30.86419752 = 364.19753082 30.86419752 + 11.11111111x + x2 = 364.19753082 Factor a perfect square on the left side: (x + 5.555555555)(x + 5.555555555) = 364.19753082 Calculate the square root of the right side: 19.08396004 Break this problem into two subproblems by setting (x + 5.555555555) equal to 19.08396004 and -19.08396004.Subproblem 1
x + 5.555555555 = 19.08396004 Simplifying x + 5.555555555 = 19.08396004 Reorder the terms: 5.555555555 + x = 19.08396004 Solving 5.555555555 + x = 19.08396004 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.555555555' to each side of the equation. 5.555555555 + -5.555555555 + x = 19.08396004 + -5.555555555 Combine like terms: 5.555555555 + -5.555555555 = 0.000000000 0.000000000 + x = 19.08396004 + -5.555555555 x = 19.08396004 + -5.555555555 Combine like terms: 19.08396004 + -5.555555555 = 13.528404485 x = 13.528404485 Simplifying x = 13.528404485Subproblem 2
x + 5.555555555 = -19.08396004 Simplifying x + 5.555555555 = -19.08396004 Reorder the terms: 5.555555555 + x = -19.08396004 Solving 5.555555555 + x = -19.08396004 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.555555555' to each side of the equation. 5.555555555 + -5.555555555 + x = -19.08396004 + -5.555555555 Combine like terms: 5.555555555 + -5.555555555 = 0.000000000 0.000000000 + x = -19.08396004 + -5.555555555 x = -19.08396004 + -5.555555555 Combine like terms: -19.08396004 + -5.555555555 = -24.639515595 x = -24.639515595 Simplifying x = -24.639515595Solution
The solution to the problem is based on the solutions from the subproblems. x = {13.528404485, -24.639515595}
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