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Simplifying (2x + 3)(2x + 4) = 1 Reorder the terms: (3 + 2x)(2x + 4) = 1 Reorder the terms: (3 + 2x)(4 + 2x) = 1 Multiply (3 + 2x) * (4 + 2x) (3(4 + 2x) + 2x * (4 + 2x)) = 1 ((4 * 3 + 2x * 3) + 2x * (4 + 2x)) = 1 ((12 + 6x) + 2x * (4 + 2x)) = 1 (12 + 6x + (4 * 2x + 2x * 2x)) = 1 (12 + 6x + (8x + 4x2)) = 1 Combine like terms: 6x + 8x = 14x (12 + 14x + 4x2) = 1 Solving 12 + 14x + 4x2 = 1 Solving for variable 'x'. Reorder the terms: 12 + -1 + 14x + 4x2 = 1 + -1 Combine like terms: 12 + -1 = 11 11 + 14x + 4x2 = 1 + -1 Combine like terms: 1 + -1 = 0 11 + 14x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.75 + 3.5x + x2 = 0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + 3.5x + -2.75 + x2 = 0 + -2.75 Reorder the terms: 2.75 + -2.75 + 3.5x + x2 = 0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + 3.5x + x2 = 0 + -2.75 3.5x + x2 = 0 + -2.75 Combine like terms: 0 + -2.75 = -2.75 3.5x + x2 = -2.75 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = -2.75 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = -2.75 + 3.0625 Combine like terms: -2.75 + 3.0625 = 0.3125 3.0625 + 3.5x + x2 = 0.3125 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (x + 1.75) equal to 0.559016994 and -0.559016994.Subproblem 1
x + 1.75 = 0.559016994 Simplifying x + 1.75 = 0.559016994 Reorder the terms: 1.75 + x = 0.559016994 Solving 1.75 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 0.559016994 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 0.559016994 + -1.75 x = 0.559016994 + -1.75 Combine like terms: 0.559016994 + -1.75 = -1.190983006 x = -1.190983006 Simplifying x = -1.190983006Subproblem 2
x + 1.75 = -0.559016994 Simplifying x + 1.75 = -0.559016994 Reorder the terms: 1.75 + x = -0.559016994 Solving 1.75 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -0.559016994 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -0.559016994 + -1.75 x = -0.559016994 + -1.75 Combine like terms: -0.559016994 + -1.75 = -2.309016994 x = -2.309016994 Simplifying x = -2.309016994Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.190983006, -2.309016994}
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