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Simplifying 5n2 + 10n + -10 = 0 Reorder the terms: -10 + 10n + 5n2 = 0 Solving -10 + 10n + 5n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '5'. 5(-2 + 2n + n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-2 + 2n + n2)' equal to zero and attempt to solve: Simplifying -2 + 2n + n2 = 0 Solving -2 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 2n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 2n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 2n + n2 = 0 + 2 2n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 2n + n2 = 2 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 = 2 + 1 Reorder the terms: 1 + 2n + n2 = 2 + 1 Combine like terms: 2 + 1 = 3 1 + 2n + n2 = 3 Factor a perfect square on the left side: (n + 1)(n + 1) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (n + 1) equal to 1.732050808 and -1.732050808.Subproblem 1
n + 1 = 1.732050808 Simplifying n + 1 = 1.732050808 Reorder the terms: 1 + n = 1.732050808 Solving 1 + n = 1.732050808 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 = 1.732050808 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.732050808 + -1 n = 1.732050808 + -1 Combine like terms: 1.732050808 + -1 = 0.732050808 n = 0.732050808 Simplifying n = 0.732050808Subproblem 2
n + 1 = -1.732050808 Simplifying n + 1 = -1.732050808 Reorder the terms: 1 + n = -1.732050808 Solving 1 + n = -1.732050808 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 = -1.732050808 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.732050808 + -1 n = -1.732050808 + -1 Combine like terms: -1.732050808 + -1 = -2.732050808 n = -2.732050808 Simplifying n = -2.732050808Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.732050808, -2.732050808}Solution
n = {0.732050808, -2.732050808}
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