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Simplifying (4x + 1)(7x + 3) = 180 Reorder the terms: (1 + 4x)(7x + 3) = 180 Reorder the terms: (1 + 4x)(3 + 7x) = 180 Multiply (1 + 4x) * (3 + 7x) (1(3 + 7x) + 4x * (3 + 7x)) = 180 ((3 * 1 + 7x * 1) + 4x * (3 + 7x)) = 180 ((3 + 7x) + 4x * (3 + 7x)) = 180 (3 + 7x + (3 * 4x + 7x * 4x)) = 180 (3 + 7x + (12x + 28x2)) = 180 Combine like terms: 7x + 12x = 19x (3 + 19x + 28x2) = 180 Solving 3 + 19x + 28x2 = 180 Solving for variable 'x'. Reorder the terms: 3 + -180 + 19x + 28x2 = 180 + -180 Combine like terms: 3 + -180 = -177 -177 + 19x + 28x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -177 + 19x + 28x2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -6.321428571 + 0.6785714286x + x2 = 0 Move the constant term to the right: Add '6.321428571' to each side of the equation. -6.321428571 + 0.6785714286x + 6.321428571 + x2 = 0 + 6.321428571 Reorder the terms: -6.321428571 + 6.321428571 + 0.6785714286x + x2 = 0 + 6.321428571 Combine like terms: -6.321428571 + 6.321428571 = 0.000000000 0.000000000 + 0.6785714286x + x2 = 0 + 6.321428571 0.6785714286x + x2 = 0 + 6.321428571 Combine like terms: 0 + 6.321428571 = 6.321428571 0.6785714286x + x2 = 6.321428571 The x term is 0.6785714286x. Take half its coefficient (0.3392857143). Square it (0.1151147959) and add it to both sides. Add '0.1151147959' to each side of the equation. 0.6785714286x + 0.1151147959 + x2 = 6.321428571 + 0.1151147959 Reorder the terms: 0.1151147959 + 0.6785714286x + x2 = 6.321428571 + 0.1151147959 Combine like terms: 6.321428571 + 0.1151147959 = 6.4365433669 0.1151147959 + 0.6785714286x + x2 = 6.4365433669 Factor a perfect square on the left side: (x + 0.3392857143)(x + 0.3392857143) = 6.4365433669 Calculate the square root of the right side: 2.537034365 Break this problem into two subproblems by setting (x + 0.3392857143) equal to 2.537034365 and -2.537034365.Subproblem 1
x + 0.3392857143 = 2.537034365 Simplifying x + 0.3392857143 = 2.537034365 Reorder the terms: 0.3392857143 + x = 2.537034365 Solving 0.3392857143 + x = 2.537034365 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3392857143' to each side of the equation. 0.3392857143 + -0.3392857143 + x = 2.537034365 + -0.3392857143 Combine like terms: 0.3392857143 + -0.3392857143 = 0.0000000000 0.0000000000 + x = 2.537034365 + -0.3392857143 x = 2.537034365 + -0.3392857143 Combine like terms: 2.537034365 + -0.3392857143 = 2.1977486507 x = 2.1977486507 Simplifying x = 2.1977486507Subproblem 2
x + 0.3392857143 = -2.537034365 Simplifying x + 0.3392857143 = -2.537034365 Reorder the terms: 0.3392857143 + x = -2.537034365 Solving 0.3392857143 + x = -2.537034365 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3392857143' to each side of the equation. 0.3392857143 + -0.3392857143 + x = -2.537034365 + -0.3392857143 Combine like terms: 0.3392857143 + -0.3392857143 = 0.0000000000 0.0000000000 + x = -2.537034365 + -0.3392857143 x = -2.537034365 + -0.3392857143 Combine like terms: -2.537034365 + -0.3392857143 = -2.8763200793 x = -2.8763200793 Simplifying x = -2.8763200793Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.1977486507, -2.8763200793}
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