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Simplifying 12x2 + -47x + -40 = 0 Reorder the terms: -40 + -47x + 12x2 = 0 Solving -40 + -47x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -3.333333333 + -3.916666667x + x2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + -3.916666667x + 3.333333333 + x2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + -3.916666667x + x2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + -3.916666667x + x2 = 0 + 3.333333333 -3.916666667x + x2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 -3.916666667x + x2 = 3.333333333 The x term is -3.916666667x. Take half its coefficient (-1.958333334). Square it (3.835069447) and add it to both sides. Add '3.835069447' to each side of the equation. -3.916666667x + 3.835069447 + x2 = 3.333333333 + 3.835069447 Reorder the terms: 3.835069447 + -3.916666667x + x2 = 3.333333333 + 3.835069447 Combine like terms: 3.333333333 + 3.835069447 = 7.16840278 3.835069447 + -3.916666667x + x2 = 7.16840278 Factor a perfect square on the left side: (x + -1.958333334)(x + -1.958333334) = 7.16840278 Calculate the square root of the right side: 2.677387305 Break this problem into two subproblems by setting (x + -1.958333334) equal to 2.677387305 and -2.677387305.Subproblem 1
x + -1.958333334 = 2.677387305 Simplifying x + -1.958333334 = 2.677387305 Reorder the terms: -1.958333334 + x = 2.677387305 Solving -1.958333334 + x = 2.677387305 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.958333334' to each side of the equation. -1.958333334 + 1.958333334 + x = 2.677387305 + 1.958333334 Combine like terms: -1.958333334 + 1.958333334 = 0.000000000 0.000000000 + x = 2.677387305 + 1.958333334 x = 2.677387305 + 1.958333334 Combine like terms: 2.677387305 + 1.958333334 = 4.635720639 x = 4.635720639 Simplifying x = 4.635720639Subproblem 2
x + -1.958333334 = -2.677387305 Simplifying x + -1.958333334 = -2.677387305 Reorder the terms: -1.958333334 + x = -2.677387305 Solving -1.958333334 + x = -2.677387305 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.958333334' to each side of the equation. -1.958333334 + 1.958333334 + x = -2.677387305 + 1.958333334 Combine like terms: -1.958333334 + 1.958333334 = 0.000000000 0.000000000 + x = -2.677387305 + 1.958333334 x = -2.677387305 + 1.958333334 Combine like terms: -2.677387305 + 1.958333334 = -0.719053971 x = -0.719053971 Simplifying x = -0.719053971Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.635720639, -0.719053971}
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