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Simplifying (6x + 7)(13x + 21) = 181 Reorder the terms: (7 + 6x)(13x + 21) = 181 Reorder the terms: (7 + 6x)(21 + 13x) = 181 Multiply (7 + 6x) * (21 + 13x) (7(21 + 13x) + 6x * (21 + 13x)) = 181 ((21 * 7 + 13x * 7) + 6x * (21 + 13x)) = 181 ((147 + 91x) + 6x * (21 + 13x)) = 181 (147 + 91x + (21 * 6x + 13x * 6x)) = 181 (147 + 91x + (126x + 78x2)) = 181 Combine like terms: 91x + 126x = 217x (147 + 217x + 78x2) = 181 Solving 147 + 217x + 78x2 = 181 Solving for variable 'x'. Reorder the terms: 147 + -181 + 217x + 78x2 = 181 + -181 Combine like terms: 147 + -181 = -34 -34 + 217x + 78x2 = 181 + -181 Combine like terms: 181 + -181 = 0 -34 + 217x + 78x2 = 0 Begin completing the square. Divide all terms by 78 the coefficient of the squared term: Divide each side by '78'. -0.4358974359 + 2.782051282x + x2 = 0 Move the constant term to the right: Add '0.4358974359' to each side of the equation. -0.4358974359 + 2.782051282x + 0.4358974359 + x2 = 0 + 0.4358974359 Reorder the terms: -0.4358974359 + 0.4358974359 + 2.782051282x + x2 = 0 + 0.4358974359 Combine like terms: -0.4358974359 + 0.4358974359 = 0.0000000000 0.0000000000 + 2.782051282x + x2 = 0 + 0.4358974359 2.782051282x + x2 = 0 + 0.4358974359 Combine like terms: 0 + 0.4358974359 = 0.4358974359 2.782051282x + x2 = 0.4358974359 The x term is 2.782051282x. Take half its coefficient (1.391025641). Square it (1.934952334) and add it to both sides. Add '1.934952334' to each side of the equation. 2.782051282x + 1.934952334 + x2 = 0.4358974359 + 1.934952334 Reorder the terms: 1.934952334 + 2.782051282x + x2 = 0.4358974359 + 1.934952334 Combine like terms: 0.4358974359 + 1.934952334 = 2.3708497699 1.934952334 + 2.782051282x + x2 = 2.3708497699 Factor a perfect square on the left side: (x + 1.391025641)(x + 1.391025641) = 2.3708497699 Calculate the square root of the right side: 1.5397564 Break this problem into two subproblems by setting (x + 1.391025641) equal to 1.5397564 and -1.5397564.Subproblem 1
x + 1.391025641 = 1.5397564 Simplifying x + 1.391025641 = 1.5397564 Reorder the terms: 1.391025641 + x = 1.5397564 Solving 1.391025641 + x = 1.5397564 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.391025641' to each side of the equation. 1.391025641 + -1.391025641 + x = 1.5397564 + -1.391025641 Combine like terms: 1.391025641 + -1.391025641 = 0.000000000 0.000000000 + x = 1.5397564 + -1.391025641 x = 1.5397564 + -1.391025641 Combine like terms: 1.5397564 + -1.391025641 = 0.148730759 x = 0.148730759 Simplifying x = 0.148730759Subproblem 2
x + 1.391025641 = -1.5397564 Simplifying x + 1.391025641 = -1.5397564 Reorder the terms: 1.391025641 + x = -1.5397564 Solving 1.391025641 + x = -1.5397564 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.391025641' to each side of the equation. 1.391025641 + -1.391025641 + x = -1.5397564 + -1.391025641 Combine like terms: 1.391025641 + -1.391025641 = 0.000000000 0.000000000 + x = -1.5397564 + -1.391025641 x = -1.5397564 + -1.391025641 Combine like terms: -1.5397564 + -1.391025641 = -2.930782041 x = -2.930782041 Simplifying x = -2.930782041Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.148730759, -2.930782041}
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