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Simplifying (x + 6)(5x + -1) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) Reorder the terms: (6 + x)(5x + -1) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) Reorder the terms: (6 + x)(-1 + 5x) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) Multiply (6 + x) * (-1 + 5x) (6(-1 + 5x) + x(-1 + 5x)) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) ((-1 * 6 + 5x * 6) + x(-1 + 5x)) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) ((-6 + 30x) + x(-1 + 5x)) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) (-6 + 30x + (-1 * x + 5x * x)) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) (-6 + 30x + (-1x + 5x2)) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) Combine like terms: 30x + -1x = 29x (-6 + 29x + 5x2) + (x + 7)(5x + -2) = 8(5x + -2)(5x + -1) Reorder the terms: -6 + 29x + 5x2 + (7 + x)(5x + -2) = 8(5x + -2)(5x + -1) Reorder the terms: -6 + 29x + 5x2 + (7 + x)(-2 + 5x) = 8(5x + -2)(5x + -1) Multiply (7 + x) * (-2 + 5x) -6 + 29x + 5x2 + (7(-2 + 5x) + x(-2 + 5x)) = 8(5x + -2)(5x + -1) -6 + 29x + 5x2 + ((-2 * 7 + 5x * 7) + x(-2 + 5x)) = 8(5x + -2)(5x + -1) -6 + 29x + 5x2 + ((-14 + 35x) + x(-2 + 5x)) = 8(5x + -2)(5x + -1) -6 + 29x + 5x2 + (-14 + 35x + (-2 * x + 5x * x)) = 8(5x + -2)(5x + -1) -6 + 29x + 5x2 + (-14 + 35x + (-2x + 5x2)) = 8(5x + -2)(5x + -1) Combine like terms: 35x + -2x = 33x -6 + 29x + 5x2 + (-14 + 33x + 5x2) = 8(5x + -2)(5x + -1) Reorder the terms: -6 + -14 + 29x + 33x + 5x2 + 5x2 = 8(5x + -2)(5x + -1) Combine like terms: -6 + -14 = -20 -20 + 29x + 33x + 5x2 + 5x2 = 8(5x + -2)(5x + -1) Combine like terms: 29x + 33x = 62x -20 + 62x + 5x2 + 5x2 = 8(5x + -2)(5x + -1) Combine like terms: 5x2 + 5x2 = 10x2 -20 + 62x + 10x2 = 8(5x + -2)(5x + -1) Reorder the terms: -20 + 62x + 10x2 = 8(-2 + 5x)(5x + -1) Reorder the terms: -20 + 62x + 10x2 = 8(-2 + 5x)(-1 + 5x) Multiply (-2 + 5x) * (-1 + 5x) -20 + 62x + 10x2 = 8(-2(-1 + 5x) + 5x * (-1 + 5x)) -20 + 62x + 10x2 = 8((-1 * -2 + 5x * -2) + 5x * (-1 + 5x)) -20 + 62x + 10x2 = 8((2 + -10x) + 5x * (-1 + 5x)) -20 + 62x + 10x2 = 8(2 + -10x + (-1 * 5x + 5x * 5x)) -20 + 62x + 10x2 = 8(2 + -10x + (-5x + 25x2)) Combine like terms: -10x + -5x = -15x -20 + 62x + 10x2 = 8(2 + -15x + 25x2) -20 + 62x + 10x2 = (2 * 8 + -15x * 8 + 25x2 * 8) -20 + 62x + 10x2 = (16 + -120x + 200x2) Solving -20 + 62x + 10x2 = 16 + -120x + 200x2 Solving for variable 'x'. Reorder the terms: -20 + -16 + 62x + 120x + 10x2 + -200x2 = 16 + -120x + 200x2 + -16 + 120x + -200x2 Combine like terms: -20 + -16 = -36 -36 + 62x + 120x + 10x2 + -200x2 = 16 + -120x + 200x2 + -16 + 120x + -200x2 Combine like terms: 62x + 120x = 182x -36 + 182x + 10x2 + -200x2 = 16 + -120x + 200x2 + -16 + 120x + -200x2 Combine like terms: 10x2 + -200x2 = -190x2 -36 + 182x + -190x2 = 16 + -120x + 200x2 + -16 + 120x + -200x2 Reorder the terms: -36 + 182x + -190x2 = 16 + -16 + -120x + 120x + 200x2 + -200x2 Combine like terms: 16 + -16 = 0 -36 + 182x + -190x2 = 0 + -120x + 120x + 200x2 + -200x2 -36 + 182x + -190x2 = -120x + 120x + 200x2 + -200x2 Combine like terms: -120x + 120x = 0 -36 + 182x + -190x2 = 0 + 200x2 + -200x2 -36 + 182x + -190x2 = 200x2 + -200x2 Combine like terms: 200x2 + -200x2 = 0 -36 + 182x + -190x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-18 + 91x + -95x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-18 + 91x + -95x2)' equal to zero and attempt to solve: Simplifying -18 + 91x + -95x2 = 0 Solving -18 + 91x + -95x2 = 0 Begin completing the square. Divide all terms by -95 the coefficient of the squared term: Divide each side by '-95'. 0.1894736842 + -0.9578947368x + x2 = 0 Move the constant term to the right: Add '-0.1894736842' to each side of the equation. 0.1894736842 + -0.9578947368x + -0.1894736842 + x2 = 0 + -0.1894736842 Reorder the terms: 0.1894736842 + -0.1894736842 + -0.9578947368x + x2 = 0 + -0.1894736842 Combine like terms: 0.1894736842 + -0.1894736842 = 0.0000000000 0.0000000000 + -0.9578947368x + x2 = 0 + -0.1894736842 -0.9578947368x + x2 = 0 + -0.1894736842 Combine like terms: 0 + -0.1894736842 = -0.1894736842 -0.9578947368x + x2 = -0.1894736842 The x term is -0.9578947368x. Take half its coefficient (-0.4789473684). Square it (0.2293905817) and add it to both sides. Add '0.2293905817' to each side of the equation. -0.9578947368x + 0.2293905817 + x2 = -0.1894736842 + 0.2293905817 Reorder the terms: 0.2293905817 + -0.9578947368x + x2 = -0.1894736842 + 0.2293905817 Combine like terms: -0.1894736842 + 0.2293905817 = 0.0399168975 0.2293905817 + -0.9578947368x + x2 = 0.0399168975 Factor a perfect square on the left side: (x + -0.4789473684)(x + -0.4789473684) = 0.0399168975 Calculate the square root of the right side: 0.199792136 Break this problem into two subproblems by setting (x + -0.4789473684) equal to 0.199792136 and -0.199792136.Subproblem 1
x + -0.4789473684 = 0.199792136 Simplifying x + -0.4789473684 = 0.199792136 Reorder the terms: -0.4789473684 + x = 0.199792136 Solving -0.4789473684 + x = 0.199792136 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4789473684' to each side of the equation. -0.4789473684 + 0.4789473684 + x = 0.199792136 + 0.4789473684 Combine like terms: -0.4789473684 + 0.4789473684 = 0.0000000000 0.0000000000 + x = 0.199792136 + 0.4789473684 x = 0.199792136 + 0.4789473684 Combine like terms: 0.199792136 + 0.4789473684 = 0.6787395044 x = 0.6787395044 Simplifying x = 0.6787395044Subproblem 2
x + -0.4789473684 = -0.199792136 Simplifying x + -0.4789473684 = -0.199792136 Reorder the terms: -0.4789473684 + x = -0.199792136 Solving -0.4789473684 + x = -0.199792136 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4789473684' to each side of the equation. -0.4789473684 + 0.4789473684 + x = -0.199792136 + 0.4789473684 Combine like terms: -0.4789473684 + 0.4789473684 = 0.0000000000 0.0000000000 + x = -0.199792136 + 0.4789473684 x = -0.199792136 + 0.4789473684 Combine like terms: -0.199792136 + 0.4789473684 = 0.2791552324 x = 0.2791552324 Simplifying x = 0.2791552324Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6787395044, 0.2791552324}Solution
x = {0.6787395044, 0.2791552324}
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