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Simplifying (x + 3)(x + 2) = 120 Reorder the terms: (3 + x)(x + 2) = 120 Reorder the terms: (3 + x)(2 + x) = 120 Multiply (3 + x) * (2 + x) (3(2 + x) + x(2 + x)) = 120 ((2 * 3 + x * 3) + x(2 + x)) = 120 ((6 + 3x) + x(2 + x)) = 120 (6 + 3x + (2 * x + x * x)) = 120 (6 + 3x + (2x + x2)) = 120 Combine like terms: 3x + 2x = 5x (6 + 5x + x2) = 120 Solving 6 + 5x + x2 = 120 Solving for variable 'x'. Reorder the terms: 6 + -120 + 5x + x2 = 120 + -120 Combine like terms: 6 + -120 = -114 -114 + 5x + x2 = 120 + -120 Combine like terms: 120 + -120 = 0 -114 + 5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '114' to each side of the equation. -114 + 5x + 114 + x2 = 0 + 114 Reorder the terms: -114 + 114 + 5x + x2 = 0 + 114 Combine like terms: -114 + 114 = 0 0 + 5x + x2 = 0 + 114 5x + x2 = 0 + 114 Combine like terms: 0 + 114 = 114 5x + x2 = 114 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 114 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 114 + 6.25 Combine like terms: 114 + 6.25 = 120.25 6.25 + 5x + x2 = 120.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 120.25 Calculate the square root of the right side: 10.9658561 Break this problem into two subproblems by setting (x + 2.5) equal to 10.9658561 and -10.9658561.Subproblem 1
x + 2.5 = 10.9658561 Simplifying x + 2.5 = 10.9658561 Reorder the terms: 2.5 + x = 10.9658561 Solving 2.5 + x = 10.9658561 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 10.9658561 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 10.9658561 + -2.5 x = 10.9658561 + -2.5 Combine like terms: 10.9658561 + -2.5 = 8.4658561 x = 8.4658561 Simplifying x = 8.4658561Subproblem 2
x + 2.5 = -10.9658561 Simplifying x + 2.5 = -10.9658561 Reorder the terms: 2.5 + x = -10.9658561 Solving 2.5 + x = -10.9658561 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -10.9658561 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -10.9658561 + -2.5 x = -10.9658561 + -2.5 Combine like terms: -10.9658561 + -2.5 = -13.4658561 x = -13.4658561 Simplifying x = -13.4658561Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.4658561, -13.4658561}
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