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Simplifying 3(2x + 1)(4x + -3) + 2(x + 3)(x + -5) = 0 Reorder the terms: 3(1 + 2x)(4x + -3) + 2(x + 3)(x + -5) = 0 Reorder the terms: 3(1 + 2x)(-3 + 4x) + 2(x + 3)(x + -5) = 0 Multiply (1 + 2x) * (-3 + 4x) 3(1(-3 + 4x) + 2x * (-3 + 4x)) + 2(x + 3)(x + -5) = 0 3((-3 * 1 + 4x * 1) + 2x * (-3 + 4x)) + 2(x + 3)(x + -5) = 0 3((-3 + 4x) + 2x * (-3 + 4x)) + 2(x + 3)(x + -5) = 0 3(-3 + 4x + (-3 * 2x + 4x * 2x)) + 2(x + 3)(x + -5) = 0 3(-3 + 4x + (-6x + 8x2)) + 2(x + 3)(x + -5) = 0 Combine like terms: 4x + -6x = -2x 3(-3 + -2x + 8x2) + 2(x + 3)(x + -5) = 0 (-3 * 3 + -2x * 3 + 8x2 * 3) + 2(x + 3)(x + -5) = 0 (-9 + -6x + 24x2) + 2(x + 3)(x + -5) = 0 Reorder the terms: -9 + -6x + 24x2 + 2(3 + x)(x + -5) = 0 Reorder the terms: -9 + -6x + 24x2 + 2(3 + x)(-5 + x) = 0 Multiply (3 + x) * (-5 + x) -9 + -6x + 24x2 + 2(3(-5 + x) + x(-5 + x)) = 0 -9 + -6x + 24x2 + 2((-5 * 3 + x * 3) + x(-5 + x)) = 0 -9 + -6x + 24x2 + 2((-15 + 3x) + x(-5 + x)) = 0 -9 + -6x + 24x2 + 2(-15 + 3x + (-5 * x + x * x)) = 0 -9 + -6x + 24x2 + 2(-15 + 3x + (-5x + x2)) = 0 Combine like terms: 3x + -5x = -2x -9 + -6x + 24x2 + 2(-15 + -2x + x2) = 0 -9 + -6x + 24x2 + (-15 * 2 + -2x * 2 + x2 * 2) = 0 -9 + -6x + 24x2 + (-30 + -4x + 2x2) = 0 Reorder the terms: -9 + -30 + -6x + -4x + 24x2 + 2x2 = 0 Combine like terms: -9 + -30 = -39 -39 + -6x + -4x + 24x2 + 2x2 = 0 Combine like terms: -6x + -4x = -10x -39 + -10x + 24x2 + 2x2 = 0 Combine like terms: 24x2 + 2x2 = 26x2 -39 + -10x + 26x2 = 0 Solving -39 + -10x + 26x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 26 the coefficient of the squared term: Divide each side by '26'. -1.5 + -0.3846153846x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.3846153846x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.3846153846x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.3846153846x + x2 = 0 + 1.5 -0.3846153846x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -0.3846153846x + x2 = 1.5 The x term is -0.3846153846x. Take half its coefficient (-0.1923076923). Square it (0.03698224852) and add it to both sides. Add '0.03698224852' to each side of the equation. -0.3846153846x + 0.03698224852 + x2 = 1.5 + 0.03698224852 Reorder the terms: 0.03698224852 + -0.3846153846x + x2 = 1.5 + 0.03698224852 Combine like terms: 1.5 + 0.03698224852 = 1.53698224852 0.03698224852 + -0.3846153846x + x2 = 1.53698224852 Factor a perfect square on the left side: (x + -0.1923076923)(x + -0.1923076923) = 1.53698224852 Calculate the square root of the right side: 1.239750882 Break this problem into two subproblems by setting (x + -0.1923076923) equal to 1.239750882 and -1.239750882.Subproblem 1
x + -0.1923076923 = 1.239750882 Simplifying x + -0.1923076923 = 1.239750882 Reorder the terms: -0.1923076923 + x = 1.239750882 Solving -0.1923076923 + x = 1.239750882 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1923076923' to each side of the equation. -0.1923076923 + 0.1923076923 + x = 1.239750882 + 0.1923076923 Combine like terms: -0.1923076923 + 0.1923076923 = 0.0000000000 0.0000000000 + x = 1.239750882 + 0.1923076923 x = 1.239750882 + 0.1923076923 Combine like terms: 1.239750882 + 0.1923076923 = 1.4320585743 x = 1.4320585743 Simplifying x = 1.4320585743Subproblem 2
x + -0.1923076923 = -1.239750882 Simplifying x + -0.1923076923 = -1.239750882 Reorder the terms: -0.1923076923 + x = -1.239750882 Solving -0.1923076923 + x = -1.239750882 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1923076923' to each side of the equation. -0.1923076923 + 0.1923076923 + x = -1.239750882 + 0.1923076923 Combine like terms: -0.1923076923 + 0.1923076923 = 0.0000000000 0.0000000000 + x = -1.239750882 + 0.1923076923 x = -1.239750882 + 0.1923076923 Combine like terms: -1.239750882 + 0.1923076923 = -1.0474431897 x = -1.0474431897 Simplifying x = -1.0474431897Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.4320585743, -1.0474431897}
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