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Simplifying 2((2 + x)(3 + x)) = 16 + x Multiply (2 + x) * (3 + x) 2((2(3 + x) + x(3 + x))) = 16 + x 2(((3 * 2 + x * 2) + x(3 + x))) = 16 + x 2(((6 + 2x) + x(3 + x))) = 16 + x 2((6 + 2x + (3 * x + x * x))) = 16 + x 2((6 + 2x + (3x + x2))) = 16 + x Combine like terms: 2x + 3x = 5x 2((6 + 5x + x2)) = 16 + x (6 * 2 + 5x * 2 + x2 * 2) = 16 + x (12 + 10x + 2x2) = 16 + x Solving 12 + 10x + 2x2 = 16 + x Solving for variable 'x'. Reorder the terms: 12 + -16 + 10x + -1x + 2x2 = 16 + x + -16 + -1x Combine like terms: 12 + -16 = -4 -4 + 10x + -1x + 2x2 = 16 + x + -16 + -1x Combine like terms: 10x + -1x = 9x -4 + 9x + 2x2 = 16 + x + -16 + -1x Reorder the terms: -4 + 9x + 2x2 = 16 + -16 + x + -1x Combine like terms: 16 + -16 = 0 -4 + 9x + 2x2 = 0 + x + -1x -4 + 9x + 2x2 = x + -1x Combine like terms: x + -1x = 0 -4 + 9x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + 4.5x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 4.5x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + 4.5x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 4.5x + x2 = 0 + 2 4.5x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 4.5x + x2 = 2 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 2 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 2 + 5.0625 Combine like terms: 2 + 5.0625 = 7.0625 5.0625 + 4.5x + x2 = 7.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 7.0625 Calculate the square root of the right side: 2.657536453 Break this problem into two subproblems by setting (x + 2.25) equal to 2.657536453 and -2.657536453.Subproblem 1
x + 2.25 = 2.657536453 Simplifying x + 2.25 = 2.657536453 Reorder the terms: 2.25 + x = 2.657536453 Solving 2.25 + x = 2.657536453 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.657536453 + -2.25 x = 2.657536453 + -2.25 Combine like terms: 2.657536453 + -2.25 = 0.407536453 x = 0.407536453 Simplifying x = 0.407536453Subproblem 2
x + 2.25 = -2.657536453 Simplifying x + 2.25 = -2.657536453 Reorder the terms: 2.25 + x = -2.657536453 Solving 2.25 + x = -2.657536453 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.657536453 + -2.25 x = -2.657536453 + -2.25 Combine like terms: -2.657536453 + -2.25 = -4.907536453 x = -4.907536453 Simplifying x = -4.907536453Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.407536453, -4.907536453}
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