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Simplifying (x + 2)(2x + 5) = 54 Reorder the terms: (2 + x)(2x + 5) = 54 Reorder the terms: (2 + x)(5 + 2x) = 54 Multiply (2 + x) * (5 + 2x) (2(5 + 2x) + x(5 + 2x)) = 54 ((5 * 2 + 2x * 2) + x(5 + 2x)) = 54 ((10 + 4x) + x(5 + 2x)) = 54 (10 + 4x + (5 * x + 2x * x)) = 54 (10 + 4x + (5x + 2x2)) = 54 Combine like terms: 4x + 5x = 9x (10 + 9x + 2x2) = 54 Solving 10 + 9x + 2x2 = 54 Solving for variable 'x'. Reorder the terms: 10 + -54 + 9x + 2x2 = 54 + -54 Combine like terms: 10 + -54 = -44 -44 + 9x + 2x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -44 + 9x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -22 + 4.5x + x2 = 0 Move the constant term to the right: Add '22' to each side of the equation. -22 + 4.5x + 22 + x2 = 0 + 22 Reorder the terms: -22 + 22 + 4.5x + x2 = 0 + 22 Combine like terms: -22 + 22 = 0 0 + 4.5x + x2 = 0 + 22 4.5x + x2 = 0 + 22 Combine like terms: 0 + 22 = 22 4.5x + x2 = 22 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 22 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 22 + 5.0625 Combine like terms: 22 + 5.0625 = 27.0625 5.0625 + 4.5x + x2 = 27.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 27.0625 Calculate the square root of the right side: 5.202163012 Break this problem into two subproblems by setting (x + 2.25) equal to 5.202163012 and -5.202163012.Subproblem 1
x + 2.25 = 5.202163012 Simplifying x + 2.25 = 5.202163012 Reorder the terms: 2.25 + x = 5.202163012 Solving 2.25 + x = 5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 5.202163012 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 5.202163012 + -2.25 x = 5.202163012 + -2.25 Combine like terms: 5.202163012 + -2.25 = 2.952163012 x = 2.952163012 Simplifying x = 2.952163012Subproblem 2
x + 2.25 = -5.202163012 Simplifying x + 2.25 = -5.202163012 Reorder the terms: 2.25 + x = -5.202163012 Solving 2.25 + x = -5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -5.202163012 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -5.202163012 + -2.25 x = -5.202163012 + -2.25 Combine like terms: -5.202163012 + -2.25 = -7.452163012 x = -7.452163012 Simplifying x = -7.452163012Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.952163012, -7.452163012}
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