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Simplifying (3x + 22)(7x + -26) = x Reorder the terms: (22 + 3x)(7x + -26) = x Reorder the terms: (22 + 3x)(-26 + 7x) = x Multiply (22 + 3x) * (-26 + 7x) (22(-26 + 7x) + 3x * (-26 + 7x)) = x ((-26 * 22 + 7x * 22) + 3x * (-26 + 7x)) = x ((-572 + 154x) + 3x * (-26 + 7x)) = x (-572 + 154x + (-26 * 3x + 7x * 3x)) = x (-572 + 154x + (-78x + 21x2)) = x Combine like terms: 154x + -78x = 76x (-572 + 76x + 21x2) = x Solving -572 + 76x + 21x2 = x Solving for variable 'x'. Reorder the terms: -572 + 76x + -1x + 21x2 = x + -1x Combine like terms: 76x + -1x = 75x -572 + 75x + 21x2 = x + -1x Combine like terms: x + -1x = 0 -572 + 75x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -27.23809524 + 3.571428571x + x2 = 0 Move the constant term to the right: Add '27.23809524' to each side of the equation. -27.23809524 + 3.571428571x + 27.23809524 + x2 = 0 + 27.23809524 Reorder the terms: -27.23809524 + 27.23809524 + 3.571428571x + x2 = 0 + 27.23809524 Combine like terms: -27.23809524 + 27.23809524 = 0.00000000 0.00000000 + 3.571428571x + x2 = 0 + 27.23809524 3.571428571x + x2 = 0 + 27.23809524 Combine like terms: 0 + 27.23809524 = 27.23809524 3.571428571x + x2 = 27.23809524 The x term is 3.571428571x. Take half its coefficient (1.785714286). Square it (3.188775511) and add it to both sides. Add '3.188775511' to each side of the equation. 3.571428571x + 3.188775511 + x2 = 27.23809524 + 3.188775511 Reorder the terms: 3.188775511 + 3.571428571x + x2 = 27.23809524 + 3.188775511 Combine like terms: 27.23809524 + 3.188775511 = 30.426870751 3.188775511 + 3.571428571x + x2 = 30.426870751 Factor a perfect square on the left side: (x + 1.785714286)(x + 1.785714286) = 30.426870751 Calculate the square root of the right side: 5.516055724 Break this problem into two subproblems by setting (x + 1.785714286) equal to 5.516055724 and -5.516055724.Subproblem 1
x + 1.785714286 = 5.516055724 Simplifying x + 1.785714286 = 5.516055724 Reorder the terms: 1.785714286 + x = 5.516055724 Solving 1.785714286 + x = 5.516055724 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + x = 5.516055724 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + x = 5.516055724 + -1.785714286 x = 5.516055724 + -1.785714286 Combine like terms: 5.516055724 + -1.785714286 = 3.730341438 x = 3.730341438 Simplifying x = 3.730341438Subproblem 2
x + 1.785714286 = -5.516055724 Simplifying x + 1.785714286 = -5.516055724 Reorder the terms: 1.785714286 + x = -5.516055724 Solving 1.785714286 + x = -5.516055724 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + x = -5.516055724 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + x = -5.516055724 + -1.785714286 x = -5.516055724 + -1.785714286 Combine like terms: -5.516055724 + -1.785714286 = -7.30177001 x = -7.30177001 Simplifying x = -7.30177001Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.730341438, -7.30177001}
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