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Simplifying (2x + 3)(2x + 4.5) = 15 Reorder the terms: (3 + 2x)(2x + 4.5) = 15 Reorder the terms: (3 + 2x)(4.5 + 2x) = 15 Multiply (3 + 2x) * (4.5 + 2x) (3(4.5 + 2x) + 2x * (4.5 + 2x)) = 15 ((4.5 * 3 + 2x * 3) + 2x * (4.5 + 2x)) = 15 ((13.5 + 6x) + 2x * (4.5 + 2x)) = 15 (13.5 + 6x + (4.5 * 2x + 2x * 2x)) = 15 (13.5 + 6x + (9x + 4x2)) = 15 Combine like terms: 6x + 9x = 15x (13.5 + 15x + 4x2) = 15 Solving 13.5 + 15x + 4x2 = 15 Solving for variable 'x'. Reorder the terms: 13.5 + -15 + 15x + 4x2 = 15 + -15 Combine like terms: 13.5 + -15 = -1.5 -1.5 + 15x + 4x2 = 15 + -15 Combine like terms: 15 + -15 = 0 -1.5 + 15x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.375 + 3.75x + x2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + 3.75x + 0.375 + x2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + 3.75x + x2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + 3.75x + x2 = 0 + 0.375 3.75x + x2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 3.75x + x2 = 0.375 The x term is 3.75x. Take half its coefficient (1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. 3.75x + 3.515625 + x2 = 0.375 + 3.515625 Reorder the terms: 3.515625 + 3.75x + x2 = 0.375 + 3.515625 Combine like terms: 0.375 + 3.515625 = 3.890625 3.515625 + 3.75x + x2 = 3.890625 Factor a perfect square on the left side: (x + 1.875)(x + 1.875) = 3.890625 Calculate the square root of the right side: 1.97246673 Break this problem into two subproblems by setting (x + 1.875) equal to 1.97246673 and -1.97246673.Subproblem 1
x + 1.875 = 1.97246673 Simplifying x + 1.875 = 1.97246673 Reorder the terms: 1.875 + x = 1.97246673 Solving 1.875 + x = 1.97246673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + x = 1.97246673 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + x = 1.97246673 + -1.875 x = 1.97246673 + -1.875 Combine like terms: 1.97246673 + -1.875 = 0.09746673 x = 0.09746673 Simplifying x = 0.09746673Subproblem 2
x + 1.875 = -1.97246673 Simplifying x + 1.875 = -1.97246673 Reorder the terms: 1.875 + x = -1.97246673 Solving 1.875 + x = -1.97246673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + x = -1.97246673 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + x = -1.97246673 + -1.875 x = -1.97246673 + -1.875 Combine like terms: -1.97246673 + -1.875 = -3.84746673 x = -3.84746673 Simplifying x = -3.84746673Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.09746673, -3.84746673}
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