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Simplifying 6x(3x + 1) = 1 Reorder the terms: 6x(1 + 3x) = 1 (1 * 6x + 3x * 6x) = 1 (6x + 18x2) = 1 Solving 6x + 18x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 6x + 18x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 6x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.05555555556 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.05555555556' to each side of the equation. -0.05555555556 + 0.3333333333x + 0.05555555556 + x2 = 0 + 0.05555555556 Reorder the terms: -0.05555555556 + 0.05555555556 + 0.3333333333x + x2 = 0 + 0.05555555556 Combine like terms: -0.05555555556 + 0.05555555556 = 0.00000000000 0.00000000000 + 0.3333333333x + x2 = 0 + 0.05555555556 0.3333333333x + x2 = 0 + 0.05555555556 Combine like terms: 0 + 0.05555555556 = 0.05555555556 0.3333333333x + x2 = 0.05555555556 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 0.05555555556 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 0.05555555556 + 0.02777777779 Combine like terms: 0.05555555556 + 0.02777777779 = 0.08333333335 0.02777777779 + 0.3333333333x + x2 = 0.08333333335 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 0.08333333335 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 0.288675135 and -0.288675135.Subproblem 1
x + 0.1666666667 = 0.288675135 Simplifying x + 0.1666666667 = 0.288675135 Reorder the terms: 0.1666666667 + x = 0.288675135 Solving 0.1666666667 + x = 0.288675135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 0.288675135 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 0.288675135 + -0.1666666667 x = 0.288675135 + -0.1666666667 Combine like terms: 0.288675135 + -0.1666666667 = 0.1220084683 x = 0.1220084683 Simplifying x = 0.1220084683Subproblem 2
x + 0.1666666667 = -0.288675135 Simplifying x + 0.1666666667 = -0.288675135 Reorder the terms: 0.1666666667 + x = -0.288675135 Solving 0.1666666667 + x = -0.288675135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -0.288675135 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -0.288675135 + -0.1666666667 x = -0.288675135 + -0.1666666667 Combine like terms: -0.288675135 + -0.1666666667 = -0.4553418017 x = -0.4553418017 Simplifying x = -0.4553418017Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.1220084683, -0.4553418017}
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