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Simplifying 7(4x + 1)(3x + 2) = 500 Reorder the terms: 7(1 + 4x)(3x + 2) = 500 Reorder the terms: 7(1 + 4x)(2 + 3x) = 500 Multiply (1 + 4x) * (2 + 3x) 7(1(2 + 3x) + 4x * (2 + 3x)) = 500 7((2 * 1 + 3x * 1) + 4x * (2 + 3x)) = 500 7((2 + 3x) + 4x * (2 + 3x)) = 500 7(2 + 3x + (2 * 4x + 3x * 4x)) = 500 7(2 + 3x + (8x + 12x2)) = 500 Combine like terms: 3x + 8x = 11x 7(2 + 11x + 12x2) = 500 (2 * 7 + 11x * 7 + 12x2 * 7) = 500 (14 + 77x + 84x2) = 500 Solving 14 + 77x + 84x2 = 500 Solving for variable 'x'. Reorder the terms: 14 + -500 + 77x + 84x2 = 500 + -500 Combine like terms: 14 + -500 = -486 -486 + 77x + 84x2 = 500 + -500 Combine like terms: 500 + -500 = 0 -486 + 77x + 84x2 = 0 Begin completing the square. Divide all terms by 84 the coefficient of the squared term: Divide each side by '84'. -5.785714286 + 0.9166666667x + x2 = 0 Move the constant term to the right: Add '5.785714286' to each side of the equation. -5.785714286 + 0.9166666667x + 5.785714286 + x2 = 0 + 5.785714286 Reorder the terms: -5.785714286 + 5.785714286 + 0.9166666667x + x2 = 0 + 5.785714286 Combine like terms: -5.785714286 + 5.785714286 = 0.000000000 0.000000000 + 0.9166666667x + x2 = 0 + 5.785714286 0.9166666667x + x2 = 0 + 5.785714286 Combine like terms: 0 + 5.785714286 = 5.785714286 0.9166666667x + x2 = 5.785714286 The x term is 0.9166666667x. Take half its coefficient (0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. 0.9166666667x + 0.2100694445 + x2 = 5.785714286 + 0.2100694445 Reorder the terms: 0.2100694445 + 0.9166666667x + x2 = 5.785714286 + 0.2100694445 Combine like terms: 5.785714286 + 0.2100694445 = 5.9957837305 0.2100694445 + 0.9166666667x + x2 = 5.9957837305 Factor a perfect square on the left side: (x + 0.4583333334)(x + 0.4583333334) = 5.9957837305 Calculate the square root of the right side: 2.448628949 Break this problem into two subproblems by setting (x + 0.4583333334) equal to 2.448628949 and -2.448628949.Subproblem 1
x + 0.4583333334 = 2.448628949 Simplifying x + 0.4583333334 = 2.448628949 Reorder the terms: 0.4583333334 + x = 2.448628949 Solving 0.4583333334 + x = 2.448628949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + x = 2.448628949 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + x = 2.448628949 + -0.4583333334 x = 2.448628949 + -0.4583333334 Combine like terms: 2.448628949 + -0.4583333334 = 1.9902956156 x = 1.9902956156 Simplifying x = 1.9902956156Subproblem 2
x + 0.4583333334 = -2.448628949 Simplifying x + 0.4583333334 = -2.448628949 Reorder the terms: 0.4583333334 + x = -2.448628949 Solving 0.4583333334 + x = -2.448628949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + x = -2.448628949 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + x = -2.448628949 + -0.4583333334 x = -2.448628949 + -0.4583333334 Combine like terms: -2.448628949 + -0.4583333334 = -2.9069622824 x = -2.9069622824 Simplifying x = -2.9069622824Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.9902956156, -2.9069622824}
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