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Simplifying -0.5x + 2.42(-0.5x2 + 3x) = 7.65 Reorder the terms: -0.5x + 2.42(3x + -0.5x2) = 7.65 -0.5x + (3x * 2.42 + -0.5x2 * 2.42) = 7.65 -0.5x + (7.26x + -1.21x2) = 7.65 Combine like terms: -0.5x + 7.26x = 6.76x 6.76x + -1.21x2 = 7.65 Solving 6.76x + -1.21x2 = 7.65 Solving for variable 'x'. Reorder the terms: -7.65 + 6.76x + -1.21x2 = 7.65 + -7.65 Combine like terms: 7.65 + -7.65 = 0.00 -7.65 + 6.76x + -1.21x2 = 0.00 Begin completing the square. Divide all terms by -1.21 the coefficient of the squared term: Divide each side by '-1.21'. 6.32231405 + -5.58677686x + x2 = 0 Move the constant term to the right: Add '-6.32231405' to each side of the equation. 6.32231405 + -5.58677686x + -6.32231405 + x2 = 0 + -6.32231405 Reorder the terms: 6.32231405 + -6.32231405 + -5.58677686x + x2 = 0 + -6.32231405 Combine like terms: 6.32231405 + -6.32231405 = 0.00000000 0.00000000 + -5.58677686x + x2 = 0 + -6.32231405 -5.58677686x + x2 = 0 + -6.32231405 Combine like terms: 0 + -6.32231405 = -6.32231405 -5.58677686x + x2 = -6.32231405 The x term is -5.58677686x. Take half its coefficient (-2.79338843). Square it (7.803018921) and add it to both sides. Add '7.803018921' to each side of the equation. -5.58677686x + 7.803018921 + x2 = -6.32231405 + 7.803018921 Reorder the terms: 7.803018921 + -5.58677686x + x2 = -6.32231405 + 7.803018921 Combine like terms: -6.32231405 + 7.803018921 = 1.480704871 7.803018921 + -5.58677686x + x2 = 1.480704871 Factor a perfect square on the left side: (x + -2.79338843)(x + -2.79338843) = 1.480704871 Calculate the square root of the right side: 1.216842172 Break this problem into two subproblems by setting (x + -2.79338843) equal to 1.216842172 and -1.216842172.Subproblem 1
x + -2.79338843 = 1.216842172 Simplifying x + -2.79338843 = 1.216842172 Reorder the terms: -2.79338843 + x = 1.216842172 Solving -2.79338843 + x = 1.216842172 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.79338843' to each side of the equation. -2.79338843 + 2.79338843 + x = 1.216842172 + 2.79338843 Combine like terms: -2.79338843 + 2.79338843 = 0.00000000 0.00000000 + x = 1.216842172 + 2.79338843 x = 1.216842172 + 2.79338843 Combine like terms: 1.216842172 + 2.79338843 = 4.010230602 x = 4.010230602 Simplifying x = 4.010230602Subproblem 2
x + -2.79338843 = -1.216842172 Simplifying x + -2.79338843 = -1.216842172 Reorder the terms: -2.79338843 + x = -1.216842172 Solving -2.79338843 + x = -1.216842172 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.79338843' to each side of the equation. -2.79338843 + 2.79338843 + x = -1.216842172 + 2.79338843 Combine like terms: -2.79338843 + 2.79338843 = 0.00000000 0.00000000 + x = -1.216842172 + 2.79338843 x = -1.216842172 + 2.79338843 Combine like terms: -1.216842172 + 2.79338843 = 1.576546258 x = 1.576546258 Simplifying x = 1.576546258Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.010230602, 1.576546258}
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