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Simplifying 7n2 + -14n + -73 = 9 Reorder the terms: -73 + -14n + 7n2 = 9 Solving -73 + -14n + 7n2 = 9 Solving for variable 'n'. Reorder the terms: -73 + -9 + -14n + 7n2 = 9 + -9 Combine like terms: -73 + -9 = -82 -82 + -14n + 7n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -82 + -14n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -11.71428571 + -2n + n2 = 0 Move the constant term to the right: Add '11.71428571' to each side of the equation. -11.71428571 + -2n + 11.71428571 + n2 = 0 + 11.71428571 Reorder the terms: -11.71428571 + 11.71428571 + -2n + n2 = 0 + 11.71428571 Combine like terms: -11.71428571 + 11.71428571 = 0.00000000 0.00000000 + -2n + n2 = 0 + 11.71428571 -2n + n2 = 0 + 11.71428571 Combine like terms: 0 + 11.71428571 = 11.71428571 -2n + n2 = 11.71428571 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 11.71428571 + 1 Reorder the terms: 1 + -2n + n2 = 11.71428571 + 1 Combine like terms: 11.71428571 + 1 = 12.71428571 1 + -2n + n2 = 12.71428571 Factor a perfect square on the left side: (n + -1)(n + -1) = 12.71428571 Calculate the square root of the right side: 3.565709706 Break this problem into two subproblems by setting (n + -1) equal to 3.565709706 and -3.565709706.Subproblem 1
n + -1 = 3.565709706 Simplifying n + -1 = 3.565709706 Reorder the terms: -1 + n = 3.565709706 Solving -1 + n = 3.565709706 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 3.565709706 + 1 Combine like terms: -1 + 1 = 0 0 + n = 3.565709706 + 1 n = 3.565709706 + 1 Combine like terms: 3.565709706 + 1 = 4.565709706 n = 4.565709706 Simplifying n = 4.565709706Subproblem 2
n + -1 = -3.565709706 Simplifying n + -1 = -3.565709706 Reorder the terms: -1 + n = -3.565709706 Solving -1 + n = -3.565709706 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -3.565709706 + 1 Combine like terms: -1 + 1 = 0 0 + n = -3.565709706 + 1 n = -3.565709706 + 1 Combine like terms: -3.565709706 + 1 = -2.565709706 n = -2.565709706 Simplifying n = -2.565709706Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.565709706, -2.565709706}
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