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Simplifying (8 + -3x)(5x + -1x) = (3x + -4)(x + -6) + -13 Combine like terms: 5x + -1x = 4x (8 + -3x)(4x) = (3x + -4)(x + -6) + -13 Remove parenthesis around (4x) (8 + -3x) * 4x = (3x + -4)(x + -6) + -13 Reorder the terms for easier multiplication: 4x(8 + -3x) = (3x + -4)(x + -6) + -13 (8 * 4x + -3x * 4x) = (3x + -4)(x + -6) + -13 (32x + -12x2) = (3x + -4)(x + -6) + -13 Reorder the terms: 32x + -12x2 = (-4 + 3x)(x + -6) + -13 Reorder the terms: 32x + -12x2 = (-4 + 3x)(-6 + x) + -13 Multiply (-4 + 3x) * (-6 + x) 32x + -12x2 = (-4(-6 + x) + 3x * (-6 + x)) + -13 32x + -12x2 = ((-6 * -4 + x * -4) + 3x * (-6 + x)) + -13 32x + -12x2 = ((24 + -4x) + 3x * (-6 + x)) + -13 32x + -12x2 = (24 + -4x + (-6 * 3x + x * 3x)) + -13 32x + -12x2 = (24 + -4x + (-18x + 3x2)) + -13 Combine like terms: -4x + -18x = -22x 32x + -12x2 = (24 + -22x + 3x2) + -13 Reorder the terms: 32x + -12x2 = 24 + -13 + -22x + 3x2 Combine like terms: 24 + -13 = 11 32x + -12x2 = 11 + -22x + 3x2 Solving 32x + -12x2 = 11 + -22x + 3x2 Solving for variable 'x'. Reorder the terms: -11 + 32x + 22x + -12x2 + -3x2 = 11 + -22x + 3x2 + -11 + 22x + -3x2 Combine like terms: 32x + 22x = 54x -11 + 54x + -12x2 + -3x2 = 11 + -22x + 3x2 + -11 + 22x + -3x2 Combine like terms: -12x2 + -3x2 = -15x2 -11 + 54x + -15x2 = 11 + -22x + 3x2 + -11 + 22x + -3x2 Reorder the terms: -11 + 54x + -15x2 = 11 + -11 + -22x + 22x + 3x2 + -3x2 Combine like terms: 11 + -11 = 0 -11 + 54x + -15x2 = 0 + -22x + 22x + 3x2 + -3x2 -11 + 54x + -15x2 = -22x + 22x + 3x2 + -3x2 Combine like terms: -22x + 22x = 0 -11 + 54x + -15x2 = 0 + 3x2 + -3x2 -11 + 54x + -15x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 -11 + 54x + -15x2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. 0.7333333333 + -3.6x + x2 = 0 Move the constant term to the right: Add '-0.7333333333' to each side of the equation. 0.7333333333 + -3.6x + -0.7333333333 + x2 = 0 + -0.7333333333 Reorder the terms: 0.7333333333 + -0.7333333333 + -3.6x + x2 = 0 + -0.7333333333 Combine like terms: 0.7333333333 + -0.7333333333 = 0.0000000000 0.0000000000 + -3.6x + x2 = 0 + -0.7333333333 -3.6x + x2 = 0 + -0.7333333333 Combine like terms: 0 + -0.7333333333 = -0.7333333333 -3.6x + x2 = -0.7333333333 The x term is -3.6x. Take half its coefficient (-1.8). Square it (3.24) and add it to both sides. Add '3.24' to each side of the equation. -3.6x + 3.24 + x2 = -0.7333333333 + 3.24 Reorder the terms: 3.24 + -3.6x + x2 = -0.7333333333 + 3.24 Combine like terms: -0.7333333333 + 3.24 = 2.5066666667 3.24 + -3.6x + x2 = 2.5066666667 Factor a perfect square on the left side: (x + -1.8)(x + -1.8) = 2.5066666667 Calculate the square root of the right side: 1.583245612 Break this problem into two subproblems by setting (x + -1.8) equal to 1.583245612 and -1.583245612.Subproblem 1
x + -1.8 = 1.583245612 Simplifying x + -1.8 = 1.583245612 Reorder the terms: -1.8 + x = 1.583245612 Solving -1.8 + x = 1.583245612 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.8' to each side of the equation. -1.8 + 1.8 + x = 1.583245612 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + x = 1.583245612 + 1.8 x = 1.583245612 + 1.8 Combine like terms: 1.583245612 + 1.8 = 3.383245612 x = 3.383245612 Simplifying x = 3.383245612Subproblem 2
x + -1.8 = -1.583245612 Simplifying x + -1.8 = -1.583245612 Reorder the terms: -1.8 + x = -1.583245612 Solving -1.8 + x = -1.583245612 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.8' to each side of the equation. -1.8 + 1.8 + x = -1.583245612 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + x = -1.583245612 + 1.8 x = -1.583245612 + 1.8 Combine like terms: -1.583245612 + 1.8 = 0.216754388 x = 0.216754388 Simplifying x = 0.216754388Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.383245612, 0.216754388}
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