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Simplifying x = 6x2 + 9x + -27 Reorder the terms: x = -27 + 9x + 6x2 Solving x = -27 + 9x + 6x2 Solving for variable 'x'. Reorder the terms: 27 + x + -9x + -6x2 = -27 + 9x + 6x2 + 27 + -9x + -6x2 Combine like terms: x + -9x = -8x 27 + -8x + -6x2 = -27 + 9x + 6x2 + 27 + -9x + -6x2 Reorder the terms: 27 + -8x + -6x2 = -27 + 27 + 9x + -9x + 6x2 + -6x2 Combine like terms: -27 + 27 = 0 27 + -8x + -6x2 = 0 + 9x + -9x + 6x2 + -6x2 27 + -8x + -6x2 = 9x + -9x + 6x2 + -6x2 Combine like terms: 9x + -9x = 0 27 + -8x + -6x2 = 0 + 6x2 + -6x2 27 + -8x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 27 + -8x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -4.5 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 1.333333333x + 4.5 + x2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 1.333333333x + x2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 1.333333333x + x2 = 0 + 4.5 1.333333333x + x2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 1.333333333x + x2 = 4.5 The x term is 1.333333333x. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333x + 0.4444444442 + x2 = 4.5 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 4.5 + 0.4444444442 Combine like terms: 4.5 + 0.4444444442 = 4.9444444442 0.4444444442 + 1.333333333x + x2 = 4.9444444442 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 4.9444444442 Calculate the square root of the right side: 2.223610677 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 2.223610677 and -2.223610677.Subproblem 1
x + 0.6666666665 = 2.223610677 Simplifying x + 0.6666666665 = 2.223610677 Reorder the terms: 0.6666666665 + x = 2.223610677 Solving 0.6666666665 + x = 2.223610677 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 2.223610677 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 2.223610677 + -0.6666666665 x = 2.223610677 + -0.6666666665 Combine like terms: 2.223610677 + -0.6666666665 = 1.5569440105 x = 1.5569440105 Simplifying x = 1.5569440105Subproblem 2
x + 0.6666666665 = -2.223610677 Simplifying x + 0.6666666665 = -2.223610677 Reorder the terms: 0.6666666665 + x = -2.223610677 Solving 0.6666666665 + x = -2.223610677 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -2.223610677 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -2.223610677 + -0.6666666665 x = -2.223610677 + -0.6666666665 Combine like terms: -2.223610677 + -0.6666666665 = -2.8902773435 x = -2.8902773435 Simplifying x = -2.8902773435Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5569440105, -2.8902773435}
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