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Simplifying 27x + 31x2 + -1 = 0 Reorder the terms: -1 + 27x + 31x2 = 0 Solving -1 + 27x + 31x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 31 the coefficient of the squared term: Divide each side by '31'. -0.03225806452 + 0.8709677419x + x2 = 0 Move the constant term to the right: Add '0.03225806452' to each side of the equation. -0.03225806452 + 0.8709677419x + 0.03225806452 + x2 = 0 + 0.03225806452 Reorder the terms: -0.03225806452 + 0.03225806452 + 0.8709677419x + x2 = 0 + 0.03225806452 Combine like terms: -0.03225806452 + 0.03225806452 = 0.00000000000 0.00000000000 + 0.8709677419x + x2 = 0 + 0.03225806452 0.8709677419x + x2 = 0 + 0.03225806452 Combine like terms: 0 + 0.03225806452 = 0.03225806452 0.8709677419x + x2 = 0.03225806452 The x term is 0.8709677419x. Take half its coefficient (0.435483871). Square it (0.1896462019) and add it to both sides. Add '0.1896462019' to each side of the equation. 0.8709677419x + 0.1896462019 + x2 = 0.03225806452 + 0.1896462019 Reorder the terms: 0.1896462019 + 0.8709677419x + x2 = 0.03225806452 + 0.1896462019 Combine like terms: 0.03225806452 + 0.1896462019 = 0.22190426642 0.1896462019 + 0.8709677419x + x2 = 0.22190426642 Factor a perfect square on the left side: (x + 0.435483871)(x + 0.435483871) = 0.22190426642 Calculate the square root of the right side: 0.471067157 Break this problem into two subproblems by setting (x + 0.435483871) equal to 0.471067157 and -0.471067157.Subproblem 1
x + 0.435483871 = 0.471067157 Simplifying x + 0.435483871 = 0.471067157 Reorder the terms: 0.435483871 + x = 0.471067157 Solving 0.435483871 + x = 0.471067157 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.435483871' to each side of the equation. 0.435483871 + -0.435483871 + x = 0.471067157 + -0.435483871 Combine like terms: 0.435483871 + -0.435483871 = 0.000000000 0.000000000 + x = 0.471067157 + -0.435483871 x = 0.471067157 + -0.435483871 Combine like terms: 0.471067157 + -0.435483871 = 0.035583286 x = 0.035583286 Simplifying x = 0.035583286Subproblem 2
x + 0.435483871 = -0.471067157 Simplifying x + 0.435483871 = -0.471067157 Reorder the terms: 0.435483871 + x = -0.471067157 Solving 0.435483871 + x = -0.471067157 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.435483871' to each side of the equation. 0.435483871 + -0.435483871 + x = -0.471067157 + -0.435483871 Combine like terms: 0.435483871 + -0.435483871 = 0.000000000 0.000000000 + x = -0.471067157 + -0.435483871 x = -0.471067157 + -0.435483871 Combine like terms: -0.471067157 + -0.435483871 = -0.906551028 x = -0.906551028 Simplifying x = -0.906551028Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.035583286, -0.906551028}
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