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Simplifying 108 = x(3x + -41) Reorder the terms: 108 = x(-41 + 3x) 108 = (-41 * x + 3x * x) 108 = (-41x + 3x2) Solving 108 = -41x + 3x2 Solving for variable 'x'. Reorder the terms: 108 + 41x + -3x2 = -41x + 41x + 3x2 + -3x2 Combine like terms: -41x + 41x = 0 108 + 41x + -3x2 = 0 + 3x2 + -3x2 108 + 41x + -3x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 108 + 41x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -36 + -13.66666667x + x2 = 0 Move the constant term to the right: Add '36' to each side of the equation. -36 + -13.66666667x + 36 + x2 = 0 + 36 Reorder the terms: -36 + 36 + -13.66666667x + x2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + -13.66666667x + x2 = 0 + 36 -13.66666667x + x2 = 0 + 36 Combine like terms: 0 + 36 = 36 -13.66666667x + x2 = 36 The x term is -13.66666667x. Take half its coefficient (-6.833333335). Square it (46.69444447) and add it to both sides. Add '46.69444447' to each side of the equation. -13.66666667x + 46.69444447 + x2 = 36 + 46.69444447 Reorder the terms: 46.69444447 + -13.66666667x + x2 = 36 + 46.69444447 Combine like terms: 36 + 46.69444447 = 82.69444447 46.69444447 + -13.66666667x + x2 = 82.69444447 Factor a perfect square on the left side: (x + -6.833333335)(x + -6.833333335) = 82.69444447 Calculate the square root of the right side: 9.093648579 Break this problem into two subproblems by setting (x + -6.833333335) equal to 9.093648579 and -9.093648579.Subproblem 1
x + -6.833333335 = 9.093648579 Simplifying x + -6.833333335 = 9.093648579 Reorder the terms: -6.833333335 + x = 9.093648579 Solving -6.833333335 + x = 9.093648579 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.833333335' to each side of the equation. -6.833333335 + 6.833333335 + x = 9.093648579 + 6.833333335 Combine like terms: -6.833333335 + 6.833333335 = 0.000000000 0.000000000 + x = 9.093648579 + 6.833333335 x = 9.093648579 + 6.833333335 Combine like terms: 9.093648579 + 6.833333335 = 15.926981914 x = 15.926981914 Simplifying x = 15.926981914Subproblem 2
x + -6.833333335 = -9.093648579 Simplifying x + -6.833333335 = -9.093648579 Reorder the terms: -6.833333335 + x = -9.093648579 Solving -6.833333335 + x = -9.093648579 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.833333335' to each side of the equation. -6.833333335 + 6.833333335 + x = -9.093648579 + 6.833333335 Combine like terms: -6.833333335 + 6.833333335 = 0.000000000 0.000000000 + x = -9.093648579 + 6.833333335 x = -9.093648579 + 6.833333335 Combine like terms: -9.093648579 + 6.833333335 = -2.260315244 x = -2.260315244 Simplifying x = -2.260315244Solution
The solution to the problem is based on the solutions from the subproblems. x = {15.926981914, -2.260315244}
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