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Simplifying 9x(3x + 8) = 22 Reorder the terms: 9x(8 + 3x) = 22 (8 * 9x + 3x * 9x) = 22 (72x + 27x2) = 22 Solving 72x + 27x2 = 22 Solving for variable 'x'. Reorder the terms: -22 + 72x + 27x2 = 22 + -22 Combine like terms: 22 + -22 = 0 -22 + 72x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.8148148148 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '0.8148148148' to each side of the equation. -0.8148148148 + 2.666666667x + 0.8148148148 + x2 = 0 + 0.8148148148 Reorder the terms: -0.8148148148 + 0.8148148148 + 2.666666667x + x2 = 0 + 0.8148148148 Combine like terms: -0.8148148148 + 0.8148148148 = 0.0000000000 0.0000000000 + 2.666666667x + x2 = 0 + 0.8148148148 2.666666667x + x2 = 0 + 0.8148148148 Combine like terms: 0 + 0.8148148148 = 0.8148148148 2.666666667x + x2 = 0.8148148148 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 = 0.8148148148 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 0.8148148148 + 1.777777780 Combine like terms: 0.8148148148 + 1.777777780 = 2.5925925948 1.777777780 + 2.666666667x + x2 = 2.5925925948 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 2.5925925948 Calculate the square root of the right side: 1.610152972 Break this problem into two subproblems by setting (x + 1.333333334) equal to 1.610152972 and -1.610152972.Subproblem 1
x + 1.333333334 = 1.610152972 Simplifying x + 1.333333334 = 1.610152972 Reorder the terms: 1.333333334 + x = 1.610152972 Solving 1.333333334 + x = 1.610152972 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 = 1.610152972 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.610152972 + -1.333333334 x = 1.610152972 + -1.333333334 Combine like terms: 1.610152972 + -1.333333334 = 0.276819638 x = 0.276819638 Simplifying x = 0.276819638Subproblem 2
x + 1.333333334 = -1.610152972 Simplifying x + 1.333333334 = -1.610152972 Reorder the terms: 1.333333334 + x = -1.610152972 Solving 1.333333334 + x = -1.610152972 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 = -1.610152972 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.610152972 + -1.333333334 x = -1.610152972 + -1.333333334 Combine like terms: -1.610152972 + -1.333333334 = -2.943486306 x = -2.943486306 Simplifying x = -2.943486306Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.276819638, -2.943486306}
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