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Simplifying (x + -2)(x + -3) = 1 Reorder the terms: (-2 + x)(x + -3) = 1 Reorder the terms: (-2 + x)(-3 + x) = 1 Multiply (-2 + x) * (-3 + x) (-2(-3 + x) + x(-3 + x)) = 1 ((-3 * -2 + x * -2) + x(-3 + x)) = 1 ((6 + -2x) + x(-3 + x)) = 1 (6 + -2x + (-3 * x + x * x)) = 1 (6 + -2x + (-3x + x2)) = 1 Combine like terms: -2x + -3x = -5x (6 + -5x + x2) = 1 Solving 6 + -5x + x2 = 1 Solving for variable 'x'. Reorder the terms: 6 + -1 + -5x + x2 = 1 + -1 Combine like terms: 6 + -1 = 5 5 + -5x + x2 = 1 + -1 Combine like terms: 1 + -1 = 0 5 + -5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + -5x + -5 + x2 = 0 + -5 Reorder the terms: 5 + -5 + -5x + x2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -5x + x2 = 0 + -5 -5x + x2 = 0 + -5 Combine like terms: 0 + -5 = -5 -5x + x2 = -5 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 = -5 + 6.25 Reorder the terms: 6.25 + -5x + x2 = -5 + 6.25 Combine like terms: -5 + 6.25 = 1.25 6.25 + -5x + x2 = 1.25 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -2.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -2.5 = 1.118033989 Simplifying x + -2.5 = 1.118033989 Reorder the terms: -2.5 + x = 1.118033989 Solving -2.5 + x = 1.118033989 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 = 1.118033989 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 1.118033989 + 2.5 x = 1.118033989 + 2.5 Combine like terms: 1.118033989 + 2.5 = 3.618033989 x = 3.618033989 Simplifying x = 3.618033989Subproblem 2
x + -2.5 = -1.118033989 Simplifying x + -2.5 = -1.118033989 Reorder the terms: -2.5 + x = -1.118033989 Solving -2.5 + x = -1.118033989 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 = -1.118033989 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -1.118033989 + 2.5 x = -1.118033989 + 2.5 Combine like terms: -1.118033989 + 2.5 = 1.381966011 x = 1.381966011 Simplifying x = 1.381966011Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.618033989, 1.381966011}
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