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Simplifying (4x + 5)(2x + 3) = 306 Reorder the terms: (5 + 4x)(2x + 3) = 306 Reorder the terms: (5 + 4x)(3 + 2x) = 306 Multiply (5 + 4x) * (3 + 2x) (5(3 + 2x) + 4x * (3 + 2x)) = 306 ((3 * 5 + 2x * 5) + 4x * (3 + 2x)) = 306 ((15 + 10x) + 4x * (3 + 2x)) = 306 (15 + 10x + (3 * 4x + 2x * 4x)) = 306 (15 + 10x + (12x + 8x2)) = 306 Combine like terms: 10x + 12x = 22x (15 + 22x + 8x2) = 306 Solving 15 + 22x + 8x2 = 306 Solving for variable 'x'. Reorder the terms: 15 + -306 + 22x + 8x2 = 306 + -306 Combine like terms: 15 + -306 = -291 -291 + 22x + 8x2 = 306 + -306 Combine like terms: 306 + -306 = 0 -291 + 22x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -36.375 + 2.75x + x2 = 0 Move the constant term to the right: Add '36.375' to each side of the equation. -36.375 + 2.75x + 36.375 + x2 = 0 + 36.375 Reorder the terms: -36.375 + 36.375 + 2.75x + x2 = 0 + 36.375 Combine like terms: -36.375 + 36.375 = 0.000 0.000 + 2.75x + x2 = 0 + 36.375 2.75x + x2 = 0 + 36.375 Combine like terms: 0 + 36.375 = 36.375 2.75x + x2 = 36.375 The x term is 2.75x. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75x + 1.890625 + x2 = 36.375 + 1.890625 Reorder the terms: 1.890625 + 2.75x + x2 = 36.375 + 1.890625 Combine like terms: 36.375 + 1.890625 = 38.265625 1.890625 + 2.75x + x2 = 38.265625 Factor a perfect square on the left side: (x + 1.375)(x + 1.375) = 38.265625 Calculate the square root of the right side: 6.185921516 Break this problem into two subproblems by setting (x + 1.375) equal to 6.185921516 and -6.185921516.Subproblem 1
x + 1.375 = 6.185921516 Simplifying x + 1.375 = 6.185921516 Reorder the terms: 1.375 + x = 6.185921516 Solving 1.375 + x = 6.185921516 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = 6.185921516 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 6.185921516 + -1.375 x = 6.185921516 + -1.375 Combine like terms: 6.185921516 + -1.375 = 4.810921516 x = 4.810921516 Simplifying x = 4.810921516Subproblem 2
x + 1.375 = -6.185921516 Simplifying x + 1.375 = -6.185921516 Reorder the terms: 1.375 + x = -6.185921516 Solving 1.375 + x = -6.185921516 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = -6.185921516 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -6.185921516 + -1.375 x = -6.185921516 + -1.375 Combine like terms: -6.185921516 + -1.375 = -7.560921516 x = -7.560921516 Simplifying x = -7.560921516Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.810921516, -7.560921516}
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