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Simplifying (2x + 1)(x + 16) = 31 Reorder the terms: (1 + 2x)(x + 16) = 31 Reorder the terms: (1 + 2x)(16 + x) = 31 Multiply (1 + 2x) * (16 + x) (1(16 + x) + 2x * (16 + x)) = 31 ((16 * 1 + x * 1) + 2x * (16 + x)) = 31 ((16 + 1x) + 2x * (16 + x)) = 31 (16 + 1x + (16 * 2x + x * 2x)) = 31 (16 + 1x + (32x + 2x2)) = 31 Combine like terms: 1x + 32x = 33x (16 + 33x + 2x2) = 31 Solving 16 + 33x + 2x2 = 31 Solving for variable 'x'. Reorder the terms: 16 + -31 + 33x + 2x2 = 31 + -31 Combine like terms: 16 + -31 = -15 -15 + 33x + 2x2 = 31 + -31 Combine like terms: 31 + -31 = 0 -15 + 33x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 16.5x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 16.5x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 16.5x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 16.5x + x2 = 0 + 7.5 16.5x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 16.5x + x2 = 7.5 The x term is 16.5x. Take half its coefficient (8.25). Square it (68.0625) and add it to both sides. Add '68.0625' to each side of the equation. 16.5x + 68.0625 + x2 = 7.5 + 68.0625 Reorder the terms: 68.0625 + 16.5x + x2 = 7.5 + 68.0625 Combine like terms: 7.5 + 68.0625 = 75.5625 68.0625 + 16.5x + x2 = 75.5625 Factor a perfect square on the left side: (x + 8.25)(x + 8.25) = 75.5625 Calculate the square root of the right side: 8.692669325 Break this problem into two subproblems by setting (x + 8.25) equal to 8.692669325 and -8.692669325.Subproblem 1
x + 8.25 = 8.692669325 Simplifying x + 8.25 = 8.692669325 Reorder the terms: 8.25 + x = 8.692669325 Solving 8.25 + x = 8.692669325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.25' to each side of the equation. 8.25 + -8.25 + x = 8.692669325 + -8.25 Combine like terms: 8.25 + -8.25 = 0.00 0.00 + x = 8.692669325 + -8.25 x = 8.692669325 + -8.25 Combine like terms: 8.692669325 + -8.25 = 0.442669325 x = 0.442669325 Simplifying x = 0.442669325Subproblem 2
x + 8.25 = -8.692669325 Simplifying x + 8.25 = -8.692669325 Reorder the terms: 8.25 + x = -8.692669325 Solving 8.25 + x = -8.692669325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.25' to each side of the equation. 8.25 + -8.25 + x = -8.692669325 + -8.25 Combine like terms: 8.25 + -8.25 = 0.00 0.00 + x = -8.692669325 + -8.25 x = -8.692669325 + -8.25 Combine like terms: -8.692669325 + -8.25 = -16.942669325 x = -16.942669325 Simplifying x = -16.942669325Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.442669325, -16.942669325}
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