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Simplifying 27x2 + -9x + 0 = 1 Reorder the terms: 0 + -9x + 27x2 = 1 Remove the zero: -9x + 27x2 = 1 Solving -9x + 27x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + -9x + 27x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -9x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.03703703704 + -0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.03703703704' to each side of the equation. -0.03703703704 + -0.3333333333x + 0.03703703704 + x2 = 0 + 0.03703703704 Reorder the terms: -0.03703703704 + 0.03703703704 + -0.3333333333x + x2 = 0 + 0.03703703704 Combine like terms: -0.03703703704 + 0.03703703704 = 0.00000000000 0.00000000000 + -0.3333333333x + x2 = 0 + 0.03703703704 -0.3333333333x + x2 = 0 + 0.03703703704 Combine like terms: 0 + 0.03703703704 = 0.03703703704 -0.3333333333x + x2 = 0.03703703704 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 = 0.03703703704 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333x + x2 = 0.03703703704 + 0.02777777779 Combine like terms: 0.03703703704 + 0.02777777779 = 0.06481481483 0.02777777779 + -0.3333333333x + x2 = 0.06481481483 Factor a perfect square on the left side: (x + -0.1666666667)(x + -0.1666666667) = 0.06481481483 Calculate the square root of the right side: 0.254587539 Break this problem into two subproblems by setting (x + -0.1666666667) equal to 0.254587539 and -0.254587539.Subproblem 1
x + -0.1666666667 = 0.254587539 Simplifying x + -0.1666666667 = 0.254587539 Reorder the terms: -0.1666666667 + x = 0.254587539 Solving -0.1666666667 + x = 0.254587539 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 = 0.254587539 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = 0.254587539 + 0.1666666667 x = 0.254587539 + 0.1666666667 Combine like terms: 0.254587539 + 0.1666666667 = 0.4212542057 x = 0.4212542057 Simplifying x = 0.4212542057Subproblem 2
x + -0.1666666667 = -0.254587539 Simplifying x + -0.1666666667 = -0.254587539 Reorder the terms: -0.1666666667 + x = -0.254587539 Solving -0.1666666667 + x = -0.254587539 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 = -0.254587539 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = -0.254587539 + 0.1666666667 x = -0.254587539 + 0.1666666667 Combine like terms: -0.254587539 + 0.1666666667 = -0.0879208723 x = -0.0879208723 Simplifying x = -0.0879208723Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.4212542057, -0.0879208723}
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