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Simplifying 10n2 + 10n + -2 = 0 Reorder the terms: -2 + 10n + 10n2 = 0 Solving -2 + 10n + 10n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 5n + 5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 5n + 5n2)' equal to zero and attempt to solve: Simplifying -1 + 5n + 5n2 = 0 Solving -1 + 5n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.2 + n + n2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + n + 0.2 + n2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + n + n2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n + n2 = 0 + 0.2 n + n2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 n + n2 = 0.2 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = 0.2 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 0.2 + 0.25 Combine like terms: 0.2 + 0.25 = 0.45 1.25 + n2 = 0.45 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.45 Calculate the square root of the right side: 0.670820393 Break this problem into two subproblems by setting (n + 0.5) equal to 0.670820393 and -0.670820393.Subproblem 1
n + 0.5 = 0.670820393 Simplifying n + 0.5 = 0.670820393 Reorder the terms: 0.5 + n = 0.670820393 Solving 0.5 + n = 0.670820393 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.670820393 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.670820393 + -0.5 n = 0.670820393 + -0.5 Combine like terms: 0.670820393 + -0.5 = 0.170820393 n = 0.170820393 Simplifying n = 0.170820393Subproblem 2
n + 0.5 = -0.670820393 Simplifying n + 0.5 = -0.670820393 Reorder the terms: 0.5 + n = -0.670820393 Solving 0.5 + n = -0.670820393 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.670820393 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.670820393 + -0.5 n = -0.670820393 + -0.5 Combine like terms: -0.670820393 + -0.5 = -1.170820393 n = -1.170820393 Simplifying n = -1.170820393Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.170820393, -1.170820393}Solution
n = {0.170820393, -1.170820393}
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