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Simplifying n2 + -2n + -3 = 5000 Reorder the terms: -3 + -2n + n2 = 5000 Solving -3 + -2n + n2 = 5000 Solving for variable 'n'. Reorder the terms: -3 + -5000 + -2n + n2 = 5000 + -5000 Combine like terms: -3 + -5000 = -5003 -5003 + -2n + n2 = 5000 + -5000 Combine like terms: 5000 + -5000 = 0 -5003 + -2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '5003' to each side of the equation. -5003 + -2n + 5003 + n2 = 0 + 5003 Reorder the terms: -5003 + 5003 + -2n + n2 = 0 + 5003 Combine like terms: -5003 + 5003 = 0 0 + -2n + n2 = 0 + 5003 -2n + n2 = 0 + 5003 Combine like terms: 0 + 5003 = 5003 -2n + n2 = 5003 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 = 5003 + 1 Reorder the terms: 1 + -2n + n2 = 5003 + 1 Combine like terms: 5003 + 1 = 5004 1 + -2n + n2 = 5004 Factor a perfect square on the left side: (n + -1)(n + -1) = 5004 Calculate the square root of the right side: 70.738956735 Break this problem into two subproblems by setting (n + -1) equal to 70.738956735 and -70.738956735.Subproblem 1
n + -1 = 70.738956735 Simplifying n + -1 = 70.738956735 Reorder the terms: -1 + n = 70.738956735 Solving -1 + n = 70.738956735 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 = 70.738956735 + 1 Combine like terms: -1 + 1 = 0 0 + n = 70.738956735 + 1 n = 70.738956735 + 1 Combine like terms: 70.738956735 + 1 = 71.738956735 n = 71.738956735 Simplifying n = 71.738956735Subproblem 2
n + -1 = -70.738956735 Simplifying n + -1 = -70.738956735 Reorder the terms: -1 + n = -70.738956735 Solving -1 + n = -70.738956735 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 = -70.738956735 + 1 Combine like terms: -1 + 1 = 0 0 + n = -70.738956735 + 1 n = -70.738956735 + 1 Combine like terms: -70.738956735 + 1 = -69.738956735 n = -69.738956735 Simplifying n = -69.738956735Solution
The solution to the problem is based on the solutions from the subproblems. n = {71.738956735, -69.738956735}
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