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Simplifying (7x + 10)(26x + 5) = 180 Reorder the terms: (10 + 7x)(26x + 5) = 180 Reorder the terms: (10 + 7x)(5 + 26x) = 180 Multiply (10 + 7x) * (5 + 26x) (10(5 + 26x) + 7x * (5 + 26x)) = 180 ((5 * 10 + 26x * 10) + 7x * (5 + 26x)) = 180 ((50 + 260x) + 7x * (5 + 26x)) = 180 (50 + 260x + (5 * 7x + 26x * 7x)) = 180 (50 + 260x + (35x + 182x2)) = 180 Combine like terms: 260x + 35x = 295x (50 + 295x + 182x2) = 180 Solving 50 + 295x + 182x2 = 180 Solving for variable 'x'. Reorder the terms: 50 + -180 + 295x + 182x2 = 180 + -180 Combine like terms: 50 + -180 = -130 -130 + 295x + 182x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -130 + 295x + 182x2 = 0 Begin completing the square. Divide all terms by 182 the coefficient of the squared term: Divide each side by '182'. -0.7142857143 + 1.620879121x + x2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 1.620879121x + 0.7142857143 + x2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 1.620879121x + x2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 1.620879121x + x2 = 0 + 0.7142857143 1.620879121x + x2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 1.620879121x + x2 = 0.7142857143 The x term is 1.620879121x. Take half its coefficient (0.8104395605). Square it (0.6568122812) and add it to both sides. Add '0.6568122812' to each side of the equation. 1.620879121x + 0.6568122812 + x2 = 0.7142857143 + 0.6568122812 Reorder the terms: 0.6568122812 + 1.620879121x + x2 = 0.7142857143 + 0.6568122812 Combine like terms: 0.7142857143 + 0.6568122812 = 1.3710979955 0.6568122812 + 1.620879121x + x2 = 1.3710979955 Factor a perfect square on the left side: (x + 0.8104395605)(x + 0.8104395605) = 1.3710979955 Calculate the square root of the right side: 1.170938938 Break this problem into two subproblems by setting (x + 0.8104395605) equal to 1.170938938 and -1.170938938.Subproblem 1
x + 0.8104395605 = 1.170938938 Simplifying x + 0.8104395605 = 1.170938938 Reorder the terms: 0.8104395605 + x = 1.170938938 Solving 0.8104395605 + x = 1.170938938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8104395605' to each side of the equation. 0.8104395605 + -0.8104395605 + x = 1.170938938 + -0.8104395605 Combine like terms: 0.8104395605 + -0.8104395605 = 0.0000000000 0.0000000000 + x = 1.170938938 + -0.8104395605 x = 1.170938938 + -0.8104395605 Combine like terms: 1.170938938 + -0.8104395605 = 0.3604993775 x = 0.3604993775 Simplifying x = 0.3604993775Subproblem 2
x + 0.8104395605 = -1.170938938 Simplifying x + 0.8104395605 = -1.170938938 Reorder the terms: 0.8104395605 + x = -1.170938938 Solving 0.8104395605 + x = -1.170938938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8104395605' to each side of the equation. 0.8104395605 + -0.8104395605 + x = -1.170938938 + -0.8104395605 Combine like terms: 0.8104395605 + -0.8104395605 = 0.0000000000 0.0000000000 + x = -1.170938938 + -0.8104395605 x = -1.170938938 + -0.8104395605 Combine like terms: -1.170938938 + -0.8104395605 = -1.9813784985 x = -1.9813784985 Simplifying x = -1.9813784985Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3604993775, -1.9813784985}
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