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Simplifying 7nr + r = 10n2r Solving 7nr + r = 10n2r Solving for variable 'n'. Reorder the terms: 7nr + -10n2r + r = 10n2r + -10n2r Combine like terms: 10n2r + -10n2r = 0 7nr + -10n2r + r = 0 Factor out the Greatest Common Factor (GCF), 'r'. r(7n + -10n2 + 1) = 0Subproblem 1
Set the factor 'r' equal to zero and attempt to solve: Simplifying r = 0 Solving r = 0 Move all terms containing n to the left, all other terms to the right. Add '-1r' to each side of the equation. r + -1r = 0 + -1r Remove the zero: 0 = -1r Simplifying 0 = -1r The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(7n + -10n2 + 1)' equal to zero and attempt to solve: Simplifying 7n + -10n2 + 1 = 0 Reorder the terms: 1 + 7n + -10n2 = 0 Solving 1 + 7n + -10n2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -0.1 + -0.7n + n2 = 0 Move the constant term to the right: Add '0.1' to each side of the equation. -0.1 + -0.7n + 0.1 + n2 = 0 + 0.1 Reorder the terms: -0.1 + 0.1 + -0.7n + n2 = 0 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + -0.7n + n2 = 0 + 0.1 -0.7n + n2 = 0 + 0.1 Combine like terms: 0 + 0.1 = 0.1 -0.7n + n2 = 0.1 The n term is -0.7n. Take half its coefficient (-0.35). Square it (0.1225) and add it to both sides. Add '0.1225' to each side of the equation. -0.7n + 0.1225 + n2 = 0.1 + 0.1225 Reorder the terms: 0.1225 + -0.7n + n2 = 0.1 + 0.1225 Combine like terms: 0.1 + 0.1225 = 0.2225 0.1225 + -0.7n + n2 = 0.2225 Factor a perfect square on the left side: (n + -0.35)(n + -0.35) = 0.2225 Calculate the square root of the right side: 0.471699057 Break this problem into two subproblems by setting (n + -0.35) equal to 0.471699057 and -0.471699057.Subproblem 1
n + -0.35 = 0.471699057 Simplifying n + -0.35 = 0.471699057 Reorder the terms: -0.35 + n = 0.471699057 Solving -0.35 + n = 0.471699057 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.35' to each side of the equation. -0.35 + 0.35 + n = 0.471699057 + 0.35 Combine like terms: -0.35 + 0.35 = 0.00 0.00 + n = 0.471699057 + 0.35 n = 0.471699057 + 0.35 Combine like terms: 0.471699057 + 0.35 = 0.821699057 n = 0.821699057 Simplifying n = 0.821699057Subproblem 2
n + -0.35 = -0.471699057 Simplifying n + -0.35 = -0.471699057 Reorder the terms: -0.35 + n = -0.471699057 Solving -0.35 + n = -0.471699057 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.35' to each side of the equation. -0.35 + 0.35 + n = -0.471699057 + 0.35 Combine like terms: -0.35 + 0.35 = 0.00 0.00 + n = -0.471699057 + 0.35 n = -0.471699057 + 0.35 Combine like terms: -0.471699057 + 0.35 = -0.121699057 n = -0.121699057 Simplifying n = -0.121699057Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.821699057, -0.121699057}Solution
n = {0.821699057, -0.121699057}
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