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Simplifying (7x + 22)(7x + 22) = 6 Reorder the terms: (22 + 7x)(7x + 22) = 6 Reorder the terms: (22 + 7x)(22 + 7x) = 6 Multiply (22 + 7x) * (22 + 7x) (22(22 + 7x) + 7x * (22 + 7x)) = 6 ((22 * 22 + 7x * 22) + 7x * (22 + 7x)) = 6 ((484 + 154x) + 7x * (22 + 7x)) = 6 (484 + 154x + (22 * 7x + 7x * 7x)) = 6 (484 + 154x + (154x + 49x2)) = 6 Combine like terms: 154x + 154x = 308x (484 + 308x + 49x2) = 6 Solving 484 + 308x + 49x2 = 6 Solving for variable 'x'. Reorder the terms: 484 + -6 + 308x + 49x2 = 6 + -6 Combine like terms: 484 + -6 = 478 478 + 308x + 49x2 = 6 + -6 Combine like terms: 6 + -6 = 0 478 + 308x + 49x2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. 9.755102041 + 6.285714286x + x2 = 0 Move the constant term to the right: Add '-9.755102041' to each side of the equation. 9.755102041 + 6.285714286x + -9.755102041 + x2 = 0 + -9.755102041 Reorder the terms: 9.755102041 + -9.755102041 + 6.285714286x + x2 = 0 + -9.755102041 Combine like terms: 9.755102041 + -9.755102041 = 0.000000000 0.000000000 + 6.285714286x + x2 = 0 + -9.755102041 6.285714286x + x2 = 0 + -9.755102041 Combine like terms: 0 + -9.755102041 = -9.755102041 6.285714286x + x2 = -9.755102041 The x term is 6.285714286x. Take half its coefficient (3.142857143). Square it (9.877551021) and add it to both sides. Add '9.877551021' to each side of the equation. 6.285714286x + 9.877551021 + x2 = -9.755102041 + 9.877551021 Reorder the terms: 9.877551021 + 6.285714286x + x2 = -9.755102041 + 9.877551021 Combine like terms: -9.755102041 + 9.877551021 = 0.12244898 9.877551021 + 6.285714286x + x2 = 0.12244898 Factor a perfect square on the left side: (x + 3.142857143)(x + 3.142857143) = 0.12244898 Calculate the square root of the right side: 0.349927107 Break this problem into two subproblems by setting (x + 3.142857143) equal to 0.349927107 and -0.349927107.Subproblem 1
x + 3.142857143 = 0.349927107 Simplifying x + 3.142857143 = 0.349927107 Reorder the terms: 3.142857143 + x = 0.349927107 Solving 3.142857143 + x = 0.349927107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.142857143' to each side of the equation. 3.142857143 + -3.142857143 + x = 0.349927107 + -3.142857143 Combine like terms: 3.142857143 + -3.142857143 = 0.000000000 0.000000000 + x = 0.349927107 + -3.142857143 x = 0.349927107 + -3.142857143 Combine like terms: 0.349927107 + -3.142857143 = -2.792930036 x = -2.792930036 Simplifying x = -2.792930036Subproblem 2
x + 3.142857143 = -0.349927107 Simplifying x + 3.142857143 = -0.349927107 Reorder the terms: 3.142857143 + x = -0.349927107 Solving 3.142857143 + x = -0.349927107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.142857143' to each side of the equation. 3.142857143 + -3.142857143 + x = -0.349927107 + -3.142857143 Combine like terms: 3.142857143 + -3.142857143 = 0.000000000 0.000000000 + x = -0.349927107 + -3.142857143 x = -0.349927107 + -3.142857143 Combine like terms: -0.349927107 + -3.142857143 = -3.49278425 x = -3.49278425 Simplifying x = -3.49278425Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.792930036, -3.49278425}
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