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Simplifying 0 = n + 2 + -0.28n2 Reorder the terms: 0 = 2 + n + -0.28n2 Solving 0 = 2 + n + -0.28n2 Solving for variable 'n'. Combine like terms: 0 + -2 = -2 -2 + -1n + 0.28n2 = 2 + n + -0.28n2 + -2 + -1n + 0.28n2 Reorder the terms: -2 + -1n + 0.28n2 = 2 + -2 + n + -1n + -0.28n2 + 0.28n2 Combine like terms: 2 + -2 = 0 -2 + -1n + 0.28n2 = 0 + n + -1n + -0.28n2 + 0.28n2 -2 + -1n + 0.28n2 = n + -1n + -0.28n2 + 0.28n2 Combine like terms: n + -1n = 0 -2 + -1n + 0.28n2 = 0 + -0.28n2 + 0.28n2 -2 + -1n + 0.28n2 = -0.28n2 + 0.28n2 Combine like terms: -0.28n2 + 0.28n2 = 0.00 -2 + -1n + 0.28n2 = 0.00 Begin completing the square. Divide all terms by 0.28 the coefficient of the squared term: Divide each side by '0.28'. -7.142857143 + -3.571428571n + n2 = 0 Move the constant term to the right: Add '7.142857143' to each side of the equation. -7.142857143 + -3.571428571n + 7.142857143 + n2 = 0 + 7.142857143 Reorder the terms: -7.142857143 + 7.142857143 + -3.571428571n + n2 = 0 + 7.142857143 Combine like terms: -7.142857143 + 7.142857143 = 0.000000000 0.000000000 + -3.571428571n + n2 = 0 + 7.142857143 -3.571428571n + n2 = 0 + 7.142857143 Combine like terms: 0 + 7.142857143 = 7.142857143 -3.571428571n + n2 = 7.142857143 The n term is -3.571428571n. 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. -3.571428571n + 3.188775511 + n2 = 7.142857143 + 3.188775511 Reorder the terms: 3.188775511 + -3.571428571n + n2 = 7.142857143 + 3.188775511 Combine like terms: 7.142857143 + 3.188775511 = 10.331632654 3.188775511 + -3.571428571n + n2 = 10.331632654 Factor a perfect square on the left side: (n + -1.785714286)(n + -1.785714286) = 10.331632654 Calculate the square root of the right side: 3.214285714 Break this problem into two subproblems by setting (n + -1.785714286) equal to 3.214285714 and -3.214285714.Subproblem 1
n + -1.785714286 = 3.214285714 Simplifying n + -1.785714286 = 3.214285714 Reorder the terms: -1.785714286 + n = 3.214285714 Solving -1.785714286 + n = 3.214285714 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.785714286' to each side of the equation. -1.785714286 + 1.785714286 + n = 3.214285714 + 1.785714286 Combine like terms: -1.785714286 + 1.785714286 = 0.000000000 0.000000000 + n = 3.214285714 + 1.785714286 n = 3.214285714 + 1.785714286 Combine like terms: 3.214285714 + 1.785714286 = 5 n = 5 Simplifying n = 5Subproblem 2
n + -1.785714286 = -3.214285714 Simplifying n + -1.785714286 = -3.214285714 Reorder the terms: -1.785714286 + n = -3.214285714 Solving -1.785714286 + n = -3.214285714 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.785714286' to each side of the equation. -1.785714286 + 1.785714286 + n = -3.214285714 + 1.785714286 Combine like terms: -1.785714286 + 1.785714286 = 0.000000000 0.000000000 + n = -3.214285714 + 1.785714286 n = -3.214285714 + 1.785714286 Combine like terms: -3.214285714 + 1.785714286 = -1.428571428 n = -1.428571428 Simplifying n = -1.428571428Solution
The solution to the problem is based on the solutions from the subproblems. n = {5, -1.428571428}
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