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Simplifying (x + 2)(2x + 5) = 1228 Reorder the terms: (2 + x)(2x + 5) = 1228 Reorder the terms: (2 + x)(5 + 2x) = 1228 Multiply (2 + x) * (5 + 2x) (2(5 + 2x) + x(5 + 2x)) = 1228 ((5 * 2 + 2x * 2) + x(5 + 2x)) = 1228 ((10 + 4x) + x(5 + 2x)) = 1228 (10 + 4x + (5 * x + 2x * x)) = 1228 (10 + 4x + (5x + 2x2)) = 1228 Combine like terms: 4x + 5x = 9x (10 + 9x + 2x2) = 1228 Solving 10 + 9x + 2x2 = 1228 Solving for variable 'x'. Reorder the terms: 10 + -1228 + 9x + 2x2 = 1228 + -1228 Combine like terms: 10 + -1228 = -1218 -1218 + 9x + 2x2 = 1228 + -1228 Combine like terms: 1228 + -1228 = 0 -1218 + 9x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -609 + 4.5x + x2 = 0 Move the constant term to the right: Add '609' to each side of the equation. -609 + 4.5x + 609 + x2 = 0 + 609 Reorder the terms: -609 + 609 + 4.5x + x2 = 0 + 609 Combine like terms: -609 + 609 = 0 0 + 4.5x + x2 = 0 + 609 4.5x + x2 = 0 + 609 Combine like terms: 0 + 609 = 609 4.5x + x2 = 609 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 = 609 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 609 + 5.0625 Combine like terms: 609 + 5.0625 = 614.0625 5.0625 + 4.5x + x2 = 614.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 614.0625 Calculate the square root of the right side: 24.780284502 Break this problem into two subproblems by setting (x + 2.25) equal to 24.780284502 and -24.780284502.Subproblem 1
x + 2.25 = 24.780284502 Simplifying x + 2.25 = 24.780284502 Reorder the terms: 2.25 + x = 24.780284502 Solving 2.25 + x = 24.780284502 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 = 24.780284502 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 24.780284502 + -2.25 x = 24.780284502 + -2.25 Combine like terms: 24.780284502 + -2.25 = 22.530284502 x = 22.530284502 Simplifying x = 22.530284502Subproblem 2
x + 2.25 = -24.780284502 Simplifying x + 2.25 = -24.780284502 Reorder the terms: 2.25 + x = -24.780284502 Solving 2.25 + x = -24.780284502 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 = -24.780284502 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -24.780284502 + -2.25 x = -24.780284502 + -2.25 Combine like terms: -24.780284502 + -2.25 = -27.030284502 x = -27.030284502 Simplifying x = -27.030284502Solution
The solution to the problem is based on the solutions from the subproblems. x = {22.530284502, -27.030284502}
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