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Simplifying (2x + 1)(x + 1) = 54 Reorder the terms: (1 + 2x)(x + 1) = 54 Reorder the terms: (1 + 2x)(1 + x) = 54 Multiply (1 + 2x) * (1 + x) (1(1 + x) + 2x * (1 + x)) = 54 ((1 * 1 + x * 1) + 2x * (1 + x)) = 54 ((1 + 1x) + 2x * (1 + x)) = 54 (1 + 1x + (1 * 2x + x * 2x)) = 54 (1 + 1x + (2x + 2x2)) = 54 Combine like terms: 1x + 2x = 3x (1 + 3x + 2x2) = 54 Solving 1 + 3x + 2x2 = 54 Solving for variable 'x'. Reorder the terms: 1 + -54 + 3x + 2x2 = 54 + -54 Combine like terms: 1 + -54 = -53 -53 + 3x + 2x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -53 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -26.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '26.5' to each side of the equation. -26.5 + 1.5x + 26.5 + x2 = 0 + 26.5 Reorder the terms: -26.5 + 26.5 + 1.5x + x2 = 0 + 26.5 Combine like terms: -26.5 + 26.5 = 0.0 0.0 + 1.5x + x2 = 0 + 26.5 1.5x + x2 = 0 + 26.5 Combine like terms: 0 + 26.5 = 26.5 1.5x + x2 = 26.5 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 = 26.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 26.5 + 0.5625 Combine like terms: 26.5 + 0.5625 = 27.0625 0.5625 + 1.5x + x2 = 27.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 27.0625 Calculate the square root of the right side: 5.202163012 Break this problem into two subproblems by setting (x + 0.75) equal to 5.202163012 and -5.202163012.Subproblem 1
x + 0.75 = 5.202163012 Simplifying x + 0.75 = 5.202163012 Reorder the terms: 0.75 + x = 5.202163012 Solving 0.75 + x = 5.202163012 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 = 5.202163012 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 5.202163012 + -0.75 x = 5.202163012 + -0.75 Combine like terms: 5.202163012 + -0.75 = 4.452163012 x = 4.452163012 Simplifying x = 4.452163012Subproblem 2
x + 0.75 = -5.202163012 Simplifying x + 0.75 = -5.202163012 Reorder the terms: 0.75 + x = -5.202163012 Solving 0.75 + x = -5.202163012 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 = -5.202163012 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -5.202163012 + -0.75 x = -5.202163012 + -0.75 Combine like terms: -5.202163012 + -0.75 = -5.952163012 x = -5.952163012 Simplifying x = -5.952163012Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.452163012, -5.952163012}
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