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Simplifying ((x + -4)(x + -3)) + -1(0.5x(x + -4)) = 0 Reorder the terms: ((-4 + x)(x + -3)) + -1(0.5x(x + -4)) = 0 Reorder the terms: ((-4 + x)(-3 + x)) + -1(0.5x(x + -4)) = 0 Multiply (-4 + x) * (-3 + x) ((-4(-3 + x) + x(-3 + x))) + -1(0.5x(x + -4)) = 0 (((-3 * -4 + x * -4) + x(-3 + x))) + -1(0.5x(x + -4)) = 0 (((12 + -4x) + x(-3 + x))) + -1(0.5x(x + -4)) = 0 ((12 + -4x + (-3 * x + x * x))) + -1(0.5x(x + -4)) = 0 ((12 + -4x + (-3x + x2))) + -1(0.5x(x + -4)) = 0 Combine like terms: -4x + -3x = -7x ((12 + -7x + x2)) + -1(0.5x(x + -4)) = 0 (12 + -7x + x2) + -1(0.5x(x + -4)) = 0 Remove parenthesis around (12 + -7x + x2) 12 + -7x + x2 + -1(0.5x(x + -4)) = 0 Reorder the terms: 12 + -7x + x2 + -1(0.5x(-4 + x)) = 0 12 + -7x + x2 + -1((-4 * 0.5x + x * 0.5x)) = 0 12 + -7x + x2 + -1((-2x + 0.5x2)) = 0 12 + -7x + x2 + (-2x * -1 + 0.5x2 * -1) = 0 12 + -7x + x2 + (2x + -0.5x2) = 0 Reorder the terms: 12 + -7x + 2x + x2 + -0.5x2 = 0 Combine like terms: -7x + 2x = -5x 12 + -5x + x2 + -0.5x2 = 0 Combine like terms: x2 + -0.5x2 = 0.5x2 12 + -5x + 0.5x2 = 0 Solving 12 + -5x + 0.5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 24 + -10x + x2 = 0 Move the constant term to the right: Add '-24' to each side of the equation. 24 + -10x + -24 + x2 = 0 + -24 Reorder the terms: 24 + -24 + -10x + x2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + -10x + x2 = 0 + -24 -10x + x2 = 0 + -24 Combine like terms: 0 + -24 = -24 -10x + x2 = -24 The x term is -10x. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10x + 25 + x2 = -24 + 25 Reorder the terms: 25 + -10x + x2 = -24 + 25 Combine like terms: -24 + 25 = 1 25 + -10x + x2 = 1 Factor a perfect square on the left side: (x + -5)(x + -5) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (x + -5) equal to 1 and -1.Subproblem 1
x + -5 = 1 Simplifying x + -5 = 1 Reorder the terms: -5 + x = 1 Solving -5 + x = 1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 1 + 5 Combine like terms: -5 + 5 = 0 0 + x = 1 + 5 x = 1 + 5 Combine like terms: 1 + 5 = 6 x = 6 Simplifying x = 6Subproblem 2
x + -5 = -1 Simplifying x + -5 = -1 Reorder the terms: -5 + x = -1 Solving -5 + x = -1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -1 + 5 Combine like terms: -5 + 5 = 0 0 + x = -1 + 5 x = -1 + 5 Combine like terms: -1 + 5 = 4 x = 4 Simplifying x = 4Solution
The solution to the problem is based on the solutions from the subproblems. x = {6, 4}
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