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Simplifying (x + -7)(7x + 5) = 78 Reorder the terms: (-7 + x)(7x + 5) = 78 Reorder the terms: (-7 + x)(5 + 7x) = 78 Multiply (-7 + x) * (5 + 7x) (-7(5 + 7x) + x(5 + 7x)) = 78 ((5 * -7 + 7x * -7) + x(5 + 7x)) = 78 ((-35 + -49x) + x(5 + 7x)) = 78 (-35 + -49x + (5 * x + 7x * x)) = 78 (-35 + -49x + (5x + 7x2)) = 78 Combine like terms: -49x + 5x = -44x (-35 + -44x + 7x2) = 78 Solving -35 + -44x + 7x2 = 78 Solving for variable 'x'. Reorder the terms: -35 + -78 + -44x + 7x2 = 78 + -78 Combine like terms: -35 + -78 = -113 -113 + -44x + 7x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -113 + -44x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -16.14285714 + -6.285714286x + x2 = 0 Move the constant term to the right: Add '16.14285714' to each side of the equation. -16.14285714 + -6.285714286x + 16.14285714 + x2 = 0 + 16.14285714 Reorder the terms: -16.14285714 + 16.14285714 + -6.285714286x + x2 = 0 + 16.14285714 Combine like terms: -16.14285714 + 16.14285714 = 0.00000000 0.00000000 + -6.285714286x + x2 = 0 + 16.14285714 -6.285714286x + x2 = 0 + 16.14285714 Combine like terms: 0 + 16.14285714 = 16.14285714 -6.285714286x + x2 = 16.14285714 The x term is -6.285714286x. Take half its coefficient (-3.142857143). Square it (9.877551021) and add it to both sides. Add '9.877551021' to each side of the equation. -6.285714286x + 9.877551021 + x2 = 16.14285714 + 9.877551021 Reorder the terms: 9.877551021 + -6.285714286x + x2 = 16.14285714 + 9.877551021 Combine like terms: 16.14285714 + 9.877551021 = 26.020408161 9.877551021 + -6.285714286x + x2 = 26.020408161 Factor a perfect square on the left side: (x + -3.142857143)(x + -3.142857143) = 26.020408161 Calculate the square root of the right side: 5.101020306 Break this problem into two subproblems by setting (x + -3.142857143) equal to 5.101020306 and -5.101020306.Subproblem 1
x + -3.142857143 = 5.101020306 Simplifying x + -3.142857143 = 5.101020306 Reorder the terms: -3.142857143 + x = 5.101020306 Solving -3.142857143 + x = 5.101020306 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.142857143' to each side of the equation. -3.142857143 + 3.142857143 + x = 5.101020306 + 3.142857143 Combine like terms: -3.142857143 + 3.142857143 = 0.000000000 0.000000000 + x = 5.101020306 + 3.142857143 x = 5.101020306 + 3.142857143 Combine like terms: 5.101020306 + 3.142857143 = 8.243877449 x = 8.243877449 Simplifying x = 8.243877449Subproblem 2
x + -3.142857143 = -5.101020306 Simplifying x + -3.142857143 = -5.101020306 Reorder the terms: -3.142857143 + x = -5.101020306 Solving -3.142857143 + x = -5.101020306 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.142857143' to each side of the equation. -3.142857143 + 3.142857143 + x = -5.101020306 + 3.142857143 Combine like terms: -3.142857143 + 3.142857143 = 0.000000000 0.000000000 + x = -5.101020306 + 3.142857143 x = -5.101020306 + 3.142857143 Combine like terms: -5.101020306 + 3.142857143 = -1.958163163 x = -1.958163163 Simplifying x = -1.958163163Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.243877449, -1.958163163}
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