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Simplifying 54 = (2x + 5)(x + 2) Reorder the terms: 54 = (5 + 2x)(x + 2) Reorder the terms: 54 = (5 + 2x)(2 + x) Multiply (5 + 2x) * (2 + x) 54 = (5(2 + x) + 2x * (2 + x)) 54 = ((2 * 5 + x * 5) + 2x * (2 + x)) 54 = ((10 + 5x) + 2x * (2 + x)) 54 = (10 + 5x + (2 * 2x + x * 2x)) 54 = (10 + 5x + (4x + 2x2)) Combine like terms: 5x + 4x = 9x 54 = (10 + 9x + 2x2) Solving 54 = 10 + 9x + 2x2 Solving for variable 'x'. Combine like terms: 54 + -10 = 44 44 + -9x + -2x2 = 10 + 9x + 2x2 + -10 + -9x + -2x2 Reorder the terms: 44 + -9x + -2x2 = 10 + -10 + 9x + -9x + 2x2 + -2x2 Combine like terms: 10 + -10 = 0 44 + -9x + -2x2 = 0 + 9x + -9x + 2x2 + -2x2 44 + -9x + -2x2 = 9x + -9x + 2x2 + -2x2 Combine like terms: 9x + -9x = 0 44 + -9x + -2x2 = 0 + 2x2 + -2x2 44 + -9x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 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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