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Simplifying 24x2 + -82x = -33 Reorder the terms: -82x + 24x2 = -33 Solving -82x + 24x2 = -33 Solving for variable 'x'. Reorder the terms: 33 + -82x + 24x2 = -33 + 33 Combine like terms: -33 + 33 = 0 33 + -82x + 24x2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 1.375 + -3.416666667x + x2 = 0 Move the constant term to the right: Add '-1.375' to each side of the equation. 1.375 + -3.416666667x + -1.375 + x2 = 0 + -1.375 Reorder the terms: 1.375 + -1.375 + -3.416666667x + x2 = 0 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + -3.416666667x + x2 = 0 + -1.375 -3.416666667x + x2 = 0 + -1.375 Combine like terms: 0 + -1.375 = -1.375 -3.416666667x + x2 = -1.375 The x term is -3.416666667x. Take half its coefficient (-1.708333334). Square it (2.918402780) and add it to both sides. Add '2.918402780' to each side of the equation. -3.416666667x + 2.918402780 + x2 = -1.375 + 2.918402780 Reorder the terms: 2.918402780 + -3.416666667x + x2 = -1.375 + 2.918402780 Combine like terms: -1.375 + 2.918402780 = 1.54340278 2.918402780 + -3.416666667x + x2 = 1.54340278 Factor a perfect square on the left side: (x + -1.708333334)(x + -1.708333334) = 1.54340278 Calculate the square root of the right side: 1.242337627 Break this problem into two subproblems by setting (x + -1.708333334) equal to 1.242337627 and -1.242337627.Subproblem 1
x + -1.708333334 = 1.242337627 Simplifying x + -1.708333334 = 1.242337627 Reorder the terms: -1.708333334 + x = 1.242337627 Solving -1.708333334 + x = 1.242337627 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.708333334' to each side of the equation. -1.708333334 + 1.708333334 + x = 1.242337627 + 1.708333334 Combine like terms: -1.708333334 + 1.708333334 = 0.000000000 0.000000000 + x = 1.242337627 + 1.708333334 x = 1.242337627 + 1.708333334 Combine like terms: 1.242337627 + 1.708333334 = 2.950670961 x = 2.950670961 Simplifying x = 2.950670961Subproblem 2
x + -1.708333334 = -1.242337627 Simplifying x + -1.708333334 = -1.242337627 Reorder the terms: -1.708333334 + x = -1.242337627 Solving -1.708333334 + x = -1.242337627 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.708333334' to each side of the equation. -1.708333334 + 1.708333334 + x = -1.242337627 + 1.708333334 Combine like terms: -1.708333334 + 1.708333334 = 0.000000000 0.000000000 + x = -1.242337627 + 1.708333334 x = -1.242337627 + 1.708333334 Combine like terms: -1.242337627 + 1.708333334 = 0.465995707 x = 0.465995707 Simplifying x = 0.465995707Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.950670961, 0.465995707}
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