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Simplifying 9x2 + -24x + 11 = 0 Reorder the terms: 11 + -24x + 9x2 = 0 Solving 11 + -24x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1.222222222 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '-1.222222222' to each side of the equation. 1.222222222 + -2.666666667x + -1.222222222 + x2 = 0 + -1.222222222 Reorder the terms: 1.222222222 + -1.222222222 + -2.666666667x + x2 = 0 + -1.222222222 Combine like terms: 1.222222222 + -1.222222222 = 0.000000000 0.000000000 + -2.666666667x + x2 = 0 + -1.222222222 -2.666666667x + x2 = 0 + -1.222222222 Combine like terms: 0 + -1.222222222 = -1.222222222 -2.666666667x + x2 = -1.222222222 The x term is -2.666666667x. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667x + 1.777777780 + x2 = -1.222222222 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = -1.222222222 + 1.777777780 Combine like terms: -1.222222222 + 1.777777780 = 0.555555558 1.777777780 + -2.666666667x + x2 = 0.555555558 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 0.555555558 Calculate the square root of the right side: 0.745355994 Break this problem into two subproblems by setting (x + -1.333333334) equal to 0.745355994 and -0.745355994.Subproblem 1
x + -1.333333334 = 0.745355994 Simplifying x + -1.333333334 = 0.745355994 Reorder the terms: -1.333333334 + x = 0.745355994 Solving -1.333333334 + x = 0.745355994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 0.745355994 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 0.745355994 + 1.333333334 x = 0.745355994 + 1.333333334 Combine like terms: 0.745355994 + 1.333333334 = 2.078689328 x = 2.078689328 Simplifying x = 2.078689328Subproblem 2
x + -1.333333334 = -0.745355994 Simplifying x + -1.333333334 = -0.745355994 Reorder the terms: -1.333333334 + x = -0.745355994 Solving -1.333333334 + x = -0.745355994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -0.745355994 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -0.745355994 + 1.333333334 x = -0.745355994 + 1.333333334 Combine like terms: -0.745355994 + 1.333333334 = 0.58797734 x = 0.58797734 Simplifying x = 0.58797734Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.078689328, 0.58797734}
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