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Simplifying (x + 9)(6x + 7) = (7x + -4)(5x + -2) Reorder the terms: (9 + x)(6x + 7) = (7x + -4)(5x + -2) Reorder the terms: (9 + x)(7 + 6x) = (7x + -4)(5x + -2) Multiply (9 + x) * (7 + 6x) (9(7 + 6x) + x(7 + 6x)) = (7x + -4)(5x + -2) ((7 * 9 + 6x * 9) + x(7 + 6x)) = (7x + -4)(5x + -2) ((63 + 54x) + x(7 + 6x)) = (7x + -4)(5x + -2) (63 + 54x + (7 * x + 6x * x)) = (7x + -4)(5x + -2) (63 + 54x + (7x + 6x2)) = (7x + -4)(5x + -2) Combine like terms: 54x + 7x = 61x (63 + 61x + 6x2) = (7x + -4)(5x + -2) Reorder the terms: 63 + 61x + 6x2 = (-4 + 7x)(5x + -2) Reorder the terms: 63 + 61x + 6x2 = (-4 + 7x)(-2 + 5x) Multiply (-4 + 7x) * (-2 + 5x) 63 + 61x + 6x2 = (-4(-2 + 5x) + 7x * (-2 + 5x)) 63 + 61x + 6x2 = ((-2 * -4 + 5x * -4) + 7x * (-2 + 5x)) 63 + 61x + 6x2 = ((8 + -20x) + 7x * (-2 + 5x)) 63 + 61x + 6x2 = (8 + -20x + (-2 * 7x + 5x * 7x)) 63 + 61x + 6x2 = (8 + -20x + (-14x + 35x2)) Combine like terms: -20x + -14x = -34x 63 + 61x + 6x2 = (8 + -34x + 35x2) Solving 63 + 61x + 6x2 = 8 + -34x + 35x2 Solving for variable 'x'. Reorder the terms: 63 + -8 + 61x + 34x + 6x2 + -35x2 = 8 + -34x + 35x2 + -8 + 34x + -35x2 Combine like terms: 63 + -8 = 55 55 + 61x + 34x + 6x2 + -35x2 = 8 + -34x + 35x2 + -8 + 34x + -35x2 Combine like terms: 61x + 34x = 95x 55 + 95x + 6x2 + -35x2 = 8 + -34x + 35x2 + -8 + 34x + -35x2 Combine like terms: 6x2 + -35x2 = -29x2 55 + 95x + -29x2 = 8 + -34x + 35x2 + -8 + 34x + -35x2 Reorder the terms: 55 + 95x + -29x2 = 8 + -8 + -34x + 34x + 35x2 + -35x2 Combine like terms: 8 + -8 = 0 55 + 95x + -29x2 = 0 + -34x + 34x + 35x2 + -35x2 55 + 95x + -29x2 = -34x + 34x + 35x2 + -35x2 Combine like terms: -34x + 34x = 0 55 + 95x + -29x2 = 0 + 35x2 + -35x2 55 + 95x + -29x2 = 35x2 + -35x2 Combine like terms: 35x2 + -35x2 = 0 55 + 95x + -29x2 = 0 Begin completing the square. Divide all terms by -29 the coefficient of the squared term: Divide each side by '-29'. -1.896551724 + -3.275862069x + x2 = 0 Move the constant term to the right: Add '1.896551724' to each side of the equation. -1.896551724 + -3.275862069x + 1.896551724 + x2 = 0 + 1.896551724 Reorder the terms: -1.896551724 + 1.896551724 + -3.275862069x + x2 = 0 + 1.896551724 Combine like terms: -1.896551724 + 1.896551724 = 0.000000000 0.000000000 + -3.275862069x + x2 = 0 + 1.896551724 -3.275862069x + x2 = 0 + 1.896551724 Combine like terms: 0 + 1.896551724 = 1.896551724 -3.275862069x + x2 = 1.896551724 The x term is -3.275862069x. Take half its coefficient (-1.637931035). Square it (2.682818075) and add it to both sides. Add '2.682818075' to each side of the equation. -3.275862069x + 2.682818075 + x2 = 1.896551724 + 2.682818075 Reorder the terms: 2.682818075 + -3.275862069x + x2 = 1.896551724 + 2.682818075 Combine like terms: 1.896551724 + 2.682818075 = 4.579369799 2.682818075 + -3.275862069x + x2 = 4.579369799 Factor a perfect square on the left side: (x + -1.637931035)(x + -1.637931035) = 4.579369799 Calculate the square root of the right side: 2.139946214 Break this problem into two subproblems by setting (x + -1.637931035) equal to 2.139946214 and -2.139946214.Subproblem 1
x + -1.637931035 = 2.139946214 Simplifying x + -1.637931035 = 2.139946214 Reorder the terms: -1.637931035 + x = 2.139946214 Solving -1.637931035 + x = 2.139946214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.637931035' to each side of the equation. -1.637931035 + 1.637931035 + x = 2.139946214 + 1.637931035 Combine like terms: -1.637931035 + 1.637931035 = 0.000000000 0.000000000 + x = 2.139946214 + 1.637931035 x = 2.139946214 + 1.637931035 Combine like terms: 2.139946214 + 1.637931035 = 3.777877249 x = 3.777877249 Simplifying x = 3.777877249Subproblem 2
x + -1.637931035 = -2.139946214 Simplifying x + -1.637931035 = -2.139946214 Reorder the terms: -1.637931035 + x = -2.139946214 Solving -1.637931035 + x = -2.139946214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.637931035' to each side of the equation. -1.637931035 + 1.637931035 + x = -2.139946214 + 1.637931035 Combine like terms: -1.637931035 + 1.637931035 = 0.000000000 0.000000000 + x = -2.139946214 + 1.637931035 x = -2.139946214 + 1.637931035 Combine like terms: -2.139946214 + 1.637931035 = -0.502015179 x = -0.502015179 Simplifying x = -0.502015179Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.777877249, -0.502015179}
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