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Simplifying (7x + 1)(18x + 4) = 31 Reorder the terms: (1 + 7x)(18x + 4) = 31 Reorder the terms: (1 + 7x)(4 + 18x) = 31 Multiply (1 + 7x) * (4 + 18x) (1(4 + 18x) + 7x * (4 + 18x)) = 31 ((4 * 1 + 18x * 1) + 7x * (4 + 18x)) = 31 ((4 + 18x) + 7x * (4 + 18x)) = 31 (4 + 18x + (4 * 7x + 18x * 7x)) = 31 (4 + 18x + (28x + 126x2)) = 31 Combine like terms: 18x + 28x = 46x (4 + 46x + 126x2) = 31 Solving 4 + 46x + 126x2 = 31 Solving for variable 'x'. Reorder the terms: 4 + -31 + 46x + 126x2 = 31 + -31 Combine like terms: 4 + -31 = -27 -27 + 46x + 126x2 = 31 + -31 Combine like terms: 31 + -31 = 0 -27 + 46x + 126x2 = 0 Begin completing the square. Divide all terms by 126 the coefficient of the squared term: Divide each side by '126'. -0.2142857143 + 0.3650793651x + x2 = 0 Move the constant term to the right: Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.3650793651x + 0.2142857143 + x2 = 0 + 0.2142857143 Reorder the terms: -0.2142857143 + 0.2142857143 + 0.3650793651x + x2 = 0 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + 0.3650793651x + x2 = 0 + 0.2142857143 0.3650793651x + x2 = 0 + 0.2142857143 Combine like terms: 0 + 0.2142857143 = 0.2142857143 0.3650793651x + x2 = 0.2142857143 The x term is 0.3650793651x. Take half its coefficient (0.1825396826). Square it (0.03332073572) and add it to both sides. Add '0.03332073572' to each side of the equation. 0.3650793651x + 0.03332073572 + x2 = 0.2142857143 + 0.03332073572 Reorder the terms: 0.03332073572 + 0.3650793651x + x2 = 0.2142857143 + 0.03332073572 Combine like terms: 0.2142857143 + 0.03332073572 = 0.24760645002 0.03332073572 + 0.3650793651x + x2 = 0.24760645002 Factor a perfect square on the left side: (x + 0.1825396826)(x + 0.1825396826) = 0.24760645002 Calculate the square root of the right side: 0.497600693 Break this problem into two subproblems by setting (x + 0.1825396826) equal to 0.497600693 and -0.497600693.Subproblem 1
x + 0.1825396826 = 0.497600693 Simplifying x + 0.1825396826 = 0.497600693 Reorder the terms: 0.1825396826 + x = 0.497600693 Solving 0.1825396826 + x = 0.497600693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1825396826' to each side of the equation. 0.1825396826 + -0.1825396826 + x = 0.497600693 + -0.1825396826 Combine like terms: 0.1825396826 + -0.1825396826 = 0.0000000000 0.0000000000 + x = 0.497600693 + -0.1825396826 x = 0.497600693 + -0.1825396826 Combine like terms: 0.497600693 + -0.1825396826 = 0.3150610104 x = 0.3150610104 Simplifying x = 0.3150610104Subproblem 2
x + 0.1825396826 = -0.497600693 Simplifying x + 0.1825396826 = -0.497600693 Reorder the terms: 0.1825396826 + x = -0.497600693 Solving 0.1825396826 + x = -0.497600693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1825396826' to each side of the equation. 0.1825396826 + -0.1825396826 + x = -0.497600693 + -0.1825396826 Combine like terms: 0.1825396826 + -0.1825396826 = 0.0000000000 0.0000000000 + x = -0.497600693 + -0.1825396826 x = -0.497600693 + -0.1825396826 Combine like terms: -0.497600693 + -0.1825396826 = -0.6801403756 x = -0.6801403756 Simplifying x = -0.6801403756Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3150610104, -0.6801403756}
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