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Simplifying (4x + 5)(2x + 3) = 60 Reorder the terms: (5 + 4x)(2x + 3) = 60 Reorder the terms: (5 + 4x)(3 + 2x) = 60 Multiply (5 + 4x) * (3 + 2x) (5(3 + 2x) + 4x * (3 + 2x)) = 60 ((3 * 5 + 2x * 5) + 4x * (3 + 2x)) = 60 ((15 + 10x) + 4x * (3 + 2x)) = 60 (15 + 10x + (3 * 4x + 2x * 4x)) = 60 (15 + 10x + (12x + 8x2)) = 60 Combine like terms: 10x + 12x = 22x (15 + 22x + 8x2) = 60 Solving 15 + 22x + 8x2 = 60 Solving for variable 'x'. Reorder the terms: 15 + -60 + 22x + 8x2 = 60 + -60 Combine like terms: 15 + -60 = -45 -45 + 22x + 8x2 = 60 + -60 Combine like terms: 60 + -60 = 0 -45 + 22x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -5.625 + 2.75x + x2 = 0 Move the constant term to the right: Add '5.625' to each side of the equation. -5.625 + 2.75x + 5.625 + x2 = 0 + 5.625 Reorder the terms: -5.625 + 5.625 + 2.75x + x2 = 0 + 5.625 Combine like terms: -5.625 + 5.625 = 0.000 0.000 + 2.75x + x2 = 0 + 5.625 2.75x + x2 = 0 + 5.625 Combine like terms: 0 + 5.625 = 5.625 2.75x + x2 = 5.625 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 = 5.625 + 1.890625 Reorder the terms: 1.890625 + 2.75x + x2 = 5.625 + 1.890625 Combine like terms: 5.625 + 1.890625 = 7.515625 1.890625 + 2.75x + x2 = 7.515625 Factor a perfect square on the left side: (x + 1.375)(x + 1.375) = 7.515625 Calculate the square root of the right side: 2.741464025 Break this problem into two subproblems by setting (x + 1.375) equal to 2.741464025 and -2.741464025.Subproblem 1
x + 1.375 = 2.741464025 Simplifying x + 1.375 = 2.741464025 Reorder the terms: 1.375 + x = 2.741464025 Solving 1.375 + x = 2.741464025 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 = 2.741464025 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 2.741464025 + -1.375 x = 2.741464025 + -1.375 Combine like terms: 2.741464025 + -1.375 = 1.366464025 x = 1.366464025 Simplifying x = 1.366464025Subproblem 2
x + 1.375 = -2.741464025 Simplifying x + 1.375 = -2.741464025 Reorder the terms: 1.375 + x = -2.741464025 Solving 1.375 + x = -2.741464025 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 = -2.741464025 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -2.741464025 + -1.375 x = -2.741464025 + -1.375 Combine like terms: -2.741464025 + -1.375 = -4.116464025 x = -4.116464025 Simplifying x = -4.116464025Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.366464025, -4.116464025}
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