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Simplifying (2x + 1)(x + 1) = 42 Reorder the terms: (1 + 2x)(x + 1) = 42 Reorder the terms: (1 + 2x)(1 + x) = 42 Multiply (1 + 2x) * (1 + x) (1(1 + x) + 2x * (1 + x)) = 42 ((1 * 1 + x * 1) + 2x * (1 + x)) = 42 ((1 + 1x) + 2x * (1 + x)) = 42 (1 + 1x + (1 * 2x + x * 2x)) = 42 (1 + 1x + (2x + 2x2)) = 42 Combine like terms: 1x + 2x = 3x (1 + 3x + 2x2) = 42 Solving 1 + 3x + 2x2 = 42 Solving for variable 'x'. Reorder the terms: 1 + -42 + 3x + 2x2 = 42 + -42 Combine like terms: 1 + -42 = -41 -41 + 3x + 2x2 = 42 + -42 Combine like terms: 42 + -42 = 0 -41 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -20.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '20.5' to each side of the equation. -20.5 + 1.5x + 20.5 + x2 = 0 + 20.5 Reorder the terms: -20.5 + 20.5 + 1.5x + x2 = 0 + 20.5 Combine like terms: -20.5 + 20.5 = 0.0 0.0 + 1.5x + x2 = 0 + 20.5 1.5x + x2 = 0 + 20.5 Combine like terms: 0 + 20.5 = 20.5 1.5x + x2 = 20.5 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 20.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 20.5 + 0.5625 Combine like terms: 20.5 + 0.5625 = 21.0625 0.5625 + 1.5x + x2 = 21.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 21.0625 Calculate the square root of the right side: 4.589389938 Break this problem into two subproblems by setting (x + 0.75) equal to 4.589389938 and -4.589389938.Subproblem 1
x + 0.75 = 4.589389938 Simplifying x + 0.75 = 4.589389938 Reorder the terms: 0.75 + x = 4.589389938 Solving 0.75 + x = 4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 4.589389938 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.589389938 + -0.75 x = 4.589389938 + -0.75 Combine like terms: 4.589389938 + -0.75 = 3.839389938 x = 3.839389938 Simplifying x = 3.839389938Subproblem 2
x + 0.75 = -4.589389938 Simplifying x + 0.75 = -4.589389938 Reorder the terms: 0.75 + x = -4.589389938 Solving 0.75 + x = -4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -4.589389938 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.589389938 + -0.75 x = -4.589389938 + -0.75 Combine like terms: -4.589389938 + -0.75 = -5.339389938 x = -5.339389938 Simplifying x = -5.339389938Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.839389938, -5.339389938}
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