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Simplifying (7x + 2)(4x + 11) = 180 Reorder the terms: (2 + 7x)(4x + 11) = 180 Reorder the terms: (2 + 7x)(11 + 4x) = 180 Multiply (2 + 7x) * (11 + 4x) (2(11 + 4x) + 7x * (11 + 4x)) = 180 ((11 * 2 + 4x * 2) + 7x * (11 + 4x)) = 180 ((22 + 8x) + 7x * (11 + 4x)) = 180 (22 + 8x + (11 * 7x + 4x * 7x)) = 180 (22 + 8x + (77x + 28x2)) = 180 Combine like terms: 8x + 77x = 85x (22 + 85x + 28x2) = 180 Solving 22 + 85x + 28x2 = 180 Solving for variable 'x'. Reorder the terms: 22 + -180 + 85x + 28x2 = 180 + -180 Combine like terms: 22 + -180 = -158 -158 + 85x + 28x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -158 + 85x + 28x2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -5.642857143 + 3.035714286x + x2 = 0 Move the constant term to the right: Add '5.642857143' to each side of the equation. -5.642857143 + 3.035714286x + 5.642857143 + x2 = 0 + 5.642857143 Reorder the terms: -5.642857143 + 5.642857143 + 3.035714286x + x2 = 0 + 5.642857143 Combine like terms: -5.642857143 + 5.642857143 = 0.000000000 0.000000000 + 3.035714286x + x2 = 0 + 5.642857143 3.035714286x + x2 = 0 + 5.642857143 Combine like terms: 0 + 5.642857143 = 5.642857143 3.035714286x + x2 = 5.642857143 The x term is 3.035714286x. Take half its coefficient (1.517857143). Square it (2.303890307) and add it to both sides. Add '2.303890307' to each side of the equation. 3.035714286x + 2.303890307 + x2 = 5.642857143 + 2.303890307 Reorder the terms: 2.303890307 + 3.035714286x + x2 = 5.642857143 + 2.303890307 Combine like terms: 5.642857143 + 2.303890307 = 7.94674745 2.303890307 + 3.035714286x + x2 = 7.94674745 Factor a perfect square on the left side: (x + 1.517857143)(x + 1.517857143) = 7.94674745 Calculate the square root of the right side: 2.818997597 Break this problem into two subproblems by setting (x + 1.517857143) equal to 2.818997597 and -2.818997597.Subproblem 1
x + 1.517857143 = 2.818997597 Simplifying x + 1.517857143 = 2.818997597 Reorder the terms: 1.517857143 + x = 2.818997597 Solving 1.517857143 + x = 2.818997597 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.517857143' to each side of the equation. 1.517857143 + -1.517857143 + x = 2.818997597 + -1.517857143 Combine like terms: 1.517857143 + -1.517857143 = 0.000000000 0.000000000 + x = 2.818997597 + -1.517857143 x = 2.818997597 + -1.517857143 Combine like terms: 2.818997597 + -1.517857143 = 1.301140454 x = 1.301140454 Simplifying x = 1.301140454Subproblem 2
x + 1.517857143 = -2.818997597 Simplifying x + 1.517857143 = -2.818997597 Reorder the terms: 1.517857143 + x = -2.818997597 Solving 1.517857143 + x = -2.818997597 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.517857143' to each side of the equation. 1.517857143 + -1.517857143 + x = -2.818997597 + -1.517857143 Combine like terms: 1.517857143 + -1.517857143 = 0.000000000 0.000000000 + x = -2.818997597 + -1.517857143 x = -2.818997597 + -1.517857143 Combine like terms: -2.818997597 + -1.517857143 = -4.33685474 x = -4.33685474 Simplifying x = -4.33685474Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.301140454, -4.33685474}
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