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Simplifying (x + 3)(x + 5) = 1 Reorder the terms: (3 + x)(x + 5) = 1 Reorder the terms: (3 + x)(5 + x) = 1 Multiply (3 + x) * (5 + x) (3(5 + x) + x(5 + x)) = 1 ((5 * 3 + x * 3) + x(5 + x)) = 1 ((15 + 3x) + x(5 + x)) = 1 (15 + 3x + (5 * x + x * x)) = 1 (15 + 3x + (5x + x2)) = 1 Combine like terms: 3x + 5x = 8x (15 + 8x + x2) = 1 Solving 15 + 8x + x2 = 1 Solving for variable 'x'. Reorder the terms: 15 + -1 + 8x + x2 = 1 + -1 Combine like terms: 15 + -1 = 14 14 + 8x + x2 = 1 + -1 Combine like terms: 1 + -1 = 0 14 + 8x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-14' to each side of the equation. 14 + 8x + -14 + x2 = 0 + -14 Reorder the terms: 14 + -14 + 8x + x2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + 8x + x2 = 0 + -14 8x + x2 = 0 + -14 Combine like terms: 0 + -14 = -14 8x + x2 = -14 The x term is 8x. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8x + 16 + x2 = -14 + 16 Reorder the terms: 16 + 8x + x2 = -14 + 16 Combine like terms: -14 + 16 = 2 16 + 8x + x2 = 2 Factor a perfect square on the left side: (x + 4)(x + 4) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (x + 4) equal to 1.414213562 and -1.414213562.Subproblem 1
x + 4 = 1.414213562 Simplifying x + 4 = 1.414213562 Reorder the terms: 4 + x = 1.414213562 Solving 4 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 1.414213562 + -4 Combine like terms: 4 + -4 = 0 0 + x = 1.414213562 + -4 x = 1.414213562 + -4 Combine like terms: 1.414213562 + -4 = -2.585786438 x = -2.585786438 Simplifying x = -2.585786438Subproblem 2
x + 4 = -1.414213562 Simplifying x + 4 = -1.414213562 Reorder the terms: 4 + x = -1.414213562 Solving 4 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -1.414213562 + -4 Combine like terms: 4 + -4 = 0 0 + x = -1.414213562 + -4 x = -1.414213562 + -4 Combine like terms: -1.414213562 + -4 = -5.414213562 x = -5.414213562 Simplifying x = -5.414213562Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.585786438, -5.414213562}
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