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Simplifying 4(3x + -1) = 5(4x + -1)(3x + -2) Reorder the terms: 4(-1 + 3x) = 5(4x + -1)(3x + -2) (-1 * 4 + 3x * 4) = 5(4x + -1)(3x + -2) (-4 + 12x) = 5(4x + -1)(3x + -2) Reorder the terms: -4 + 12x = 5(-1 + 4x)(3x + -2) Reorder the terms: -4 + 12x = 5(-1 + 4x)(-2 + 3x) Multiply (-1 + 4x) * (-2 + 3x) -4 + 12x = 5(-1(-2 + 3x) + 4x * (-2 + 3x)) -4 + 12x = 5((-2 * -1 + 3x * -1) + 4x * (-2 + 3x)) -4 + 12x = 5((2 + -3x) + 4x * (-2 + 3x)) -4 + 12x = 5(2 + -3x + (-2 * 4x + 3x * 4x)) -4 + 12x = 5(2 + -3x + (-8x + 12x2)) Combine like terms: -3x + -8x = -11x -4 + 12x = 5(2 + -11x + 12x2) -4 + 12x = (2 * 5 + -11x * 5 + 12x2 * 5) -4 + 12x = (10 + -55x + 60x2) Solving -4 + 12x = 10 + -55x + 60x2 Solving for variable 'x'. Reorder the terms: -4 + -10 + 12x + 55x + -60x2 = 10 + -55x + 60x2 + -10 + 55x + -60x2 Combine like terms: -4 + -10 = -14 -14 + 12x + 55x + -60x2 = 10 + -55x + 60x2 + -10 + 55x + -60x2 Combine like terms: 12x + 55x = 67x -14 + 67x + -60x2 = 10 + -55x + 60x2 + -10 + 55x + -60x2 Reorder the terms: -14 + 67x + -60x2 = 10 + -10 + -55x + 55x + 60x2 + -60x2 Combine like terms: 10 + -10 = 0 -14 + 67x + -60x2 = 0 + -55x + 55x + 60x2 + -60x2 -14 + 67x + -60x2 = -55x + 55x + 60x2 + -60x2 Combine like terms: -55x + 55x = 0 -14 + 67x + -60x2 = 0 + 60x2 + -60x2 -14 + 67x + -60x2 = 60x2 + -60x2 Combine like terms: 60x2 + -60x2 = 0 -14 + 67x + -60x2 = 0 Begin completing the square. Divide all terms by -60 the coefficient of the squared term: Divide each side by '-60'. 0.2333333333 + -1.116666667x + x2 = 0 Move the constant term to the right: Add '-0.2333333333' to each side of the equation. 0.2333333333 + -1.116666667x + -0.2333333333 + x2 = 0 + -0.2333333333 Reorder the terms: 0.2333333333 + -0.2333333333 + -1.116666667x + x2 = 0 + -0.2333333333 Combine like terms: 0.2333333333 + -0.2333333333 = 0.0000000000 0.0000000000 + -1.116666667x + x2 = 0 + -0.2333333333 -1.116666667x + x2 = 0 + -0.2333333333 Combine like terms: 0 + -0.2333333333 = -0.2333333333 -1.116666667x + x2 = -0.2333333333 The x term is -1.116666667x. Take half its coefficient (-0.5583333335). Square it (0.3117361113) and add it to both sides. Add '0.3117361113' to each side of the equation. -1.116666667x + 0.3117361113 + x2 = -0.2333333333 + 0.3117361113 Reorder the terms: 0.3117361113 + -1.116666667x + x2 = -0.2333333333 + 0.3117361113 Combine like terms: -0.2333333333 + 0.3117361113 = 0.078402778 0.3117361113 + -1.116666667x + x2 = 0.078402778 Factor a perfect square on the left side: (x + -0.5583333335)(x + -0.5583333335) = 0.078402778 Calculate the square root of the right side: 0.280004961 Break this problem into two subproblems by setting (x + -0.5583333335) equal to 0.280004961 and -0.280004961.Subproblem 1
x + -0.5583333335 = 0.280004961 Simplifying x + -0.5583333335 = 0.280004961 Reorder the terms: -0.5583333335 + x = 0.280004961 Solving -0.5583333335 + x = 0.280004961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5583333335' to each side of the equation. -0.5583333335 + 0.5583333335 + x = 0.280004961 + 0.5583333335 Combine like terms: -0.5583333335 + 0.5583333335 = 0.0000000000 0.0000000000 + x = 0.280004961 + 0.5583333335 x = 0.280004961 + 0.5583333335 Combine like terms: 0.280004961 + 0.5583333335 = 0.8383382945 x = 0.8383382945 Simplifying x = 0.8383382945Subproblem 2
x + -0.5583333335 = -0.280004961 Simplifying x + -0.5583333335 = -0.280004961 Reorder the terms: -0.5583333335 + x = -0.280004961 Solving -0.5583333335 + x = -0.280004961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5583333335' to each side of the equation. -0.5583333335 + 0.5583333335 + x = -0.280004961 + 0.5583333335 Combine like terms: -0.5583333335 + 0.5583333335 = 0.0000000000 0.0000000000 + x = -0.280004961 + 0.5583333335 x = -0.280004961 + 0.5583333335 Combine like terms: -0.280004961 + 0.5583333335 = 0.2783283725 x = 0.2783283725 Simplifying x = 0.2783283725Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8383382945, 0.2783283725}
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