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Simplifying (x + -2)(x + -2) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: (-2 + x)(x + -2) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: (-2 + x)(-2 + x) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Multiply (-2 + x) * (-2 + x) (-2(-2 + x) + x(-2 + x)) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 ((-2 * -2 + x * -2) + x(-2 + x)) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 ((4 + -2x) + x(-2 + x)) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 (4 + -2x + (-2 * x + x * x)) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 (4 + -2x + (-2x + x2)) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Combine like terms: -2x + -2x = -4x (4 + -4x + x2) + (3x + 1)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: 4 + -4x + x2 + (1 + 3x)(3x + 1) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: 4 + -4x + x2 + (1 + 3x)(1 + 3x) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Multiply (1 + 3x) * (1 + 3x) 4 + -4x + x2 + (1(1 + 3x) + 3x * (1 + 3x)) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + ((1 * 1 + 3x * 1) + 3x * (1 + 3x)) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + ((1 + 3x) + 3x * (1 + 3x)) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + (1 + 3x + (1 * 3x + 3x * 3x)) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + (1 + 3x + (3x + 9x2)) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Combine like terms: 3x + 3x = 6x 4 + -4x + x2 + (1 + 6x + 9x2) + -1(2x + 1)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: 4 + -4x + x2 + 1 + 6x + 9x2 + -1(1 + 2x)(2x + -1) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: 4 + -4x + x2 + 1 + 6x + 9x2 + -1(1 + 2x)(-1 + 2x) + -1(2x + 3)(2x + 3) = 0 Multiply (1 + 2x) * (-1 + 2x) 4 + -4x + x2 + 1 + 6x + 9x2 + -1(1(-1 + 2x) + 2x * (-1 + 2x)) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + -1((-1 * 1 + 2x * 1) + 2x * (-1 + 2x)) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + -1((-1 + 2x) + 2x * (-1 + 2x)) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + -1(-1 + 2x + (-1 * 2x + 2x * 2x)) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + -1(-1 + 2x + (-2x + 4x2)) + -1(2x + 3)(2x + 3) = 0 Combine like terms: 2x + -2x = 0 4 + -4x + x2 + 1 + 6x + 9x2 + -1(-1 + 0 + 4x2) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + -1(-1 + 4x2) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + (-1 * -1 + 4x2 * -1) + -1(2x + 3)(2x + 3) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + (1 + -4x2) + -1(2x + 3)(2x + 3) = 0 Reorder the terms: 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1(3 + 2x)(2x + 3) = 0 Reorder the terms: 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1(3 + 2x)(3 + 2x) = 0 Multiply (3 + 2x) * (3 + 2x) 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1(3(3 + 2x) + 2x * (3 + 2x)) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1((3 * 3 + 2x * 3) + 2x * (3 + 2x)) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1((9 + 6x) + 2x * (3 + 2x)) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1(9 + 6x + (3 * 2x + 2x * 2x)) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1(9 + 6x + (6x + 4x2)) = 0 Combine like terms: 6x + 6x = 12x 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + -1(9 + 12x + 4x2) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + (9 * -1 + 12x * -1 + 4x2 * -1) = 0 4 + -4x + x2 + 1 + 6x + 9x2 + 1 + -4x2 + (-9 + -12x + -4x2) = 0 Reorder the terms: 4 + 1 + 1 + -9 + -4x + 6x + -12x + x2 + 9x2 + -4x2 + -4x2 = 0 Combine like terms: 4 + 1 = 5 5 + 1 + -9 + -4x + 6x + -12x + x2 + 9x2 + -4x2 + -4x2 = 0 Combine like terms: 5 + 1 = 6 6 + -9 + -4x + 6x + -12x + x2 + 9x2 + -4x2 + -4x2 = 0 Combine like terms: 6 + -9 = -3 -3 + -4x + 6x + -12x + x2 + 9x2 + -4x2 + -4x2 = 0 Combine like terms: -4x + 6x = 2x -3 + 2x + -12x + x2 + 9x2 + -4x2 + -4x2 = 0 Combine like terms: 2x + -12x = -10x -3 + -10x + x2 + 9x2 + -4x2 + -4x2 = 0 Combine like terms: x2 + 9x2 = 10x2 -3 + -10x + 10x2 + -4x2 + -4x2 = 0 Combine like terms: 10x2 + -4x2 = 6x2 -3 + -10x + 6x2 + -4x2 = 0 Combine like terms: 6x2 + -4x2 = 2x2 -3 + -10x + 2x2 = 0 Solving -3 + -10x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + -5x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -5x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -5x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -5x + x2 = 0 + 1.5 -5x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -5x + x2 = 1.5 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 = 1.5 + 6.25 Reorder the terms: 6.25 + -5x + x2 = 1.5 + 6.25 Combine like terms: 1.5 + 6.25 = 7.75 6.25 + -5x + x2 = 7.75 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (x + -2.5) equal to 2.783882181 and -2.783882181.Subproblem 1
x + -2.5 = 2.783882181 Simplifying x + -2.5 = 2.783882181 Reorder the terms: -2.5 + x = 2.783882181 Solving -2.5 + x = 2.783882181 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 = 2.783882181 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.783882181 + 2.5 x = 2.783882181 + 2.5 Combine like terms: 2.783882181 + 2.5 = 5.283882181 x = 5.283882181 Simplifying x = 5.283882181Subproblem 2
x + -2.5 = -2.783882181 Simplifying x + -2.5 = -2.783882181 Reorder the terms: -2.5 + x = -2.783882181 Solving -2.5 + x = -2.783882181 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 = -2.783882181 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.783882181 + 2.5 x = -2.783882181 + 2.5 Combine like terms: -2.783882181 + 2.5 = -0.283882181 x = -0.283882181 Simplifying x = -0.283882181Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.283882181, -0.283882181}
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