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Simplifying (11x + 7)(11x + 7) = 105 Reorder the terms: (7 + 11x)(11x + 7) = 105 Reorder the terms: (7 + 11x)(7 + 11x) = 105 Multiply (7 + 11x) * (7 + 11x) (7(7 + 11x) + 11x * (7 + 11x)) = 105 ((7 * 7 + 11x * 7) + 11x * (7 + 11x)) = 105 ((49 + 77x) + 11x * (7 + 11x)) = 105 (49 + 77x + (7 * 11x + 11x * 11x)) = 105 (49 + 77x + (77x + 121x2)) = 105 Combine like terms: 77x + 77x = 154x (49 + 154x + 121x2) = 105 Solving 49 + 154x + 121x2 = 105 Solving for variable 'x'. Reorder the terms: 49 + -105 + 154x + 121x2 = 105 + -105 Combine like terms: 49 + -105 = -56 -56 + 154x + 121x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -56 + 154x + 121x2 = 0 Begin completing the square. Divide all terms by 121 the coefficient of the squared term: Divide each side by '121'. -0.4628099174 + 1.272727273x + x2 = 0 Move the constant term to the right: Add '0.4628099174' to each side of the equation. -0.4628099174 + 1.272727273x + 0.4628099174 + x2 = 0 + 0.4628099174 Reorder the terms: -0.4628099174 + 0.4628099174 + 1.272727273x + x2 = 0 + 0.4628099174 Combine like terms: -0.4628099174 + 0.4628099174 = 0.0000000000 0.0000000000 + 1.272727273x + x2 = 0 + 0.4628099174 1.272727273x + x2 = 0 + 0.4628099174 Combine like terms: 0 + 0.4628099174 = 0.4628099174 1.272727273x + x2 = 0.4628099174 The x term is 1.272727273x. Take half its coefficient (0.6363636365). Square it (0.4049586779) and add it to both sides. Add '0.4049586779' to each side of the equation. 1.272727273x + 0.4049586779 + x2 = 0.4628099174 + 0.4049586779 Reorder the terms: 0.4049586779 + 1.272727273x + x2 = 0.4628099174 + 0.4049586779 Combine like terms: 0.4628099174 + 0.4049586779 = 0.8677685953 0.4049586779 + 1.272727273x + x2 = 0.8677685953 Factor a perfect square on the left side: (x + 0.6363636365)(x + 0.6363636365) = 0.8677685953 Calculate the square root of the right side: 0.931540979 Break this problem into two subproblems by setting (x + 0.6363636365) equal to 0.931540979 and -0.931540979.Subproblem 1
x + 0.6363636365 = 0.931540979 Simplifying x + 0.6363636365 = 0.931540979 Reorder the terms: 0.6363636365 + x = 0.931540979 Solving 0.6363636365 + x = 0.931540979 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6363636365' to each side of the equation. 0.6363636365 + -0.6363636365 + x = 0.931540979 + -0.6363636365 Combine like terms: 0.6363636365 + -0.6363636365 = 0.0000000000 0.0000000000 + x = 0.931540979 + -0.6363636365 x = 0.931540979 + -0.6363636365 Combine like terms: 0.931540979 + -0.6363636365 = 0.2951773425 x = 0.2951773425 Simplifying x = 0.2951773425Subproblem 2
x + 0.6363636365 = -0.931540979 Simplifying x + 0.6363636365 = -0.931540979 Reorder the terms: 0.6363636365 + x = -0.931540979 Solving 0.6363636365 + x = -0.931540979 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6363636365' to each side of the equation. 0.6363636365 + -0.6363636365 + x = -0.931540979 + -0.6363636365 Combine like terms: 0.6363636365 + -0.6363636365 = 0.0000000000 0.0000000000 + x = -0.931540979 + -0.6363636365 x = -0.931540979 + -0.6363636365 Combine like terms: -0.931540979 + -0.6363636365 = -1.5679046155 x = -1.5679046155 Simplifying x = -1.5679046155Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2951773425, -1.5679046155}
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