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Simplifying 5n2 + 10n = 20 Reorder the terms: 10n + 5n2 = 20 Solving 10n + 5n2 = 20 Solving for variable 'n'. Reorder the terms: -20 + 10n + 5n2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + 10n + 5n2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-4 + 2n + n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-4 + 2n + n2)' equal to zero and attempt to solve: Simplifying -4 + 2n + n2 = 0 Solving -4 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 2n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + 2n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2n + n2 = 0 + 4 2n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 2n + n2 = 4 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 = 4 + 1 Reorder the terms: 1 + 2n + n2 = 4 + 1 Combine like terms: 4 + 1 = 5 1 + 2n + n2 = 5 Factor a perfect square on the left side: (n + 1)(n + 1) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (n + 1) equal to 2.236067978 and -2.236067978.Subproblem 1
n + 1 = 2.236067978 Simplifying n + 1 = 2.236067978 Reorder the terms: 1 + n = 2.236067978 Solving 1 + n = 2.236067978 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 = 2.236067978 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.236067978 + -1 n = 2.236067978 + -1 Combine like terms: 2.236067978 + -1 = 1.236067978 n = 1.236067978 Simplifying n = 1.236067978Subproblem 2
n + 1 = -2.236067978 Simplifying n + 1 = -2.236067978 Reorder the terms: 1 + n = -2.236067978 Solving 1 + n = -2.236067978 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 = -2.236067978 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.236067978 + -1 n = -2.236067978 + -1 Combine like terms: -2.236067978 + -1 = -3.236067978 n = -3.236067978 Simplifying n = -3.236067978Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.236067978, -3.236067978}Solution
n = {1.236067978, -3.236067978}
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