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Simplifying (6x + 7)(5x + -3) = 180 Reorder the terms: (7 + 6x)(5x + -3) = 180 Reorder the terms: (7 + 6x)(-3 + 5x) = 180 Multiply (7 + 6x) * (-3 + 5x) (7(-3 + 5x) + 6x * (-3 + 5x)) = 180 ((-3 * 7 + 5x * 7) + 6x * (-3 + 5x)) = 180 ((-21 + 35x) + 6x * (-3 + 5x)) = 180 (-21 + 35x + (-3 * 6x + 5x * 6x)) = 180 (-21 + 35x + (-18x + 30x2)) = 180 Combine like terms: 35x + -18x = 17x (-21 + 17x + 30x2) = 180 Solving -21 + 17x + 30x2 = 180 Solving for variable 'x'. Reorder the terms: -21 + -180 + 17x + 30x2 = 180 + -180 Combine like terms: -21 + -180 = -201 -201 + 17x + 30x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -201 + 17x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -6.7 + 0.5666666667x + x2 = 0.0 Move the constant term to the right: Add '6.7' to each side of the equation. -6.7 + 0.5666666667x + 6.7 + x2 = 0.0 + 6.7 Reorder the terms: -6.7 + 6.7 + 0.5666666667x + x2 = 0.0 + 6.7 Combine like terms: -6.7 + 6.7 = 0.0 0.0 + 0.5666666667x + x2 = 0.0 + 6.7 0.5666666667x + x2 = 0.0 + 6.7 Combine like terms: 0.0 + 6.7 = 6.7 0.5666666667x + x2 = 6.7 The x term is 0.5666666667x. Take half its coefficient (0.2833333334). Square it (0.08027777782) and add it to both sides. Add '0.08027777782' to each side of the equation. 0.5666666667x + 0.08027777782 + x2 = 6.7 + 0.08027777782 Reorder the terms: 0.08027777782 + 0.5666666667x + x2 = 6.7 + 0.08027777782 Combine like terms: 6.7 + 0.08027777782 = 6.78027777782 0.08027777782 + 0.5666666667x + x2 = 6.78027777782 Factor a perfect square on the left side: (x + 0.2833333334)(x + 0.2833333334) = 6.78027777782 Calculate the square root of the right side: 2.603896653 Break this problem into two subproblems by setting (x + 0.2833333334) equal to 2.603896653 and -2.603896653.Subproblem 1
x + 0.2833333334 = 2.603896653 Simplifying x + 0.2833333334 = 2.603896653 Reorder the terms: 0.2833333334 + x = 2.603896653 Solving 0.2833333334 + x = 2.603896653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2833333334' to each side of the equation. 0.2833333334 + -0.2833333334 + x = 2.603896653 + -0.2833333334 Combine like terms: 0.2833333334 + -0.2833333334 = 0.0000000000 0.0000000000 + x = 2.603896653 + -0.2833333334 x = 2.603896653 + -0.2833333334 Combine like terms: 2.603896653 + -0.2833333334 = 2.3205633196 x = 2.3205633196 Simplifying x = 2.3205633196Subproblem 2
x + 0.2833333334 = -2.603896653 Simplifying x + 0.2833333334 = -2.603896653 Reorder the terms: 0.2833333334 + x = -2.603896653 Solving 0.2833333334 + x = -2.603896653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2833333334' to each side of the equation. 0.2833333334 + -0.2833333334 + x = -2.603896653 + -0.2833333334 Combine like terms: 0.2833333334 + -0.2833333334 = 0.0000000000 0.0000000000 + x = -2.603896653 + -0.2833333334 x = -2.603896653 + -0.2833333334 Combine like terms: -2.603896653 + -0.2833333334 = -2.8872299864 x = -2.8872299864 Simplifying x = -2.8872299864Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.3205633196, -2.8872299864}
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