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Simplifying 82n + -84 = 10n2 + -40n Reorder the terms: -84 + 82n = 10n2 + -40n Reorder the terms: -84 + 82n = -40n + 10n2 Solving -84 + 82n = -40n + 10n2 Solving for variable 'n'. Combine like terms: 82n + 40n = 122n -84 + 122n + -10n2 = -40n + 10n2 + 40n + -10n2 Reorder the terms: -84 + 122n + -10n2 = -40n + 40n + 10n2 + -10n2 Combine like terms: -40n + 40n = 0 -84 + 122n + -10n2 = 0 + 10n2 + -10n2 -84 + 122n + -10n2 = 10n2 + -10n2 Combine like terms: 10n2 + -10n2 = 0 -84 + 122n + -10n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-42 + 61n + -5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-42 + 61n + -5n2)' equal to zero and attempt to solve: Simplifying -42 + 61n + -5n2 = 0 Solving -42 + 61n + -5n2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 8.4 + -12.2n + n2 = 0 Move the constant term to the right: Add '-8.4' to each side of the equation. 8.4 + -12.2n + -8.4 + n2 = 0 + -8.4 Reorder the terms: 8.4 + -8.4 + -12.2n + n2 = 0 + -8.4 Combine like terms: 8.4 + -8.4 = 0.0 0.0 + -12.2n + n2 = 0 + -8.4 -12.2n + n2 = 0 + -8.4 Combine like terms: 0 + -8.4 = -8.4 -12.2n + n2 = -8.4 The n term is -12.2n. Take half its coefficient (-6.1). Square it (37.21) and add it to both sides. Add '37.21' to each side of the equation. -12.2n + 37.21 + n2 = -8.4 + 37.21 Reorder the terms: 37.21 + -12.2n + n2 = -8.4 + 37.21 Combine like terms: -8.4 + 37.21 = 28.81 37.21 + -12.2n + n2 = 28.81 Factor a perfect square on the left side: (n + -6.1)(n + -6.1) = 28.81 Calculate the square root of the right side: 5.36749476 Break this problem into two subproblems by setting (n + -6.1) equal to 5.36749476 and -5.36749476.Subproblem 1
n + -6.1 = 5.36749476 Simplifying n + -6.1 = 5.36749476 Reorder the terms: -6.1 + n = 5.36749476 Solving -6.1 + n = 5.36749476 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.1' to each side of the equation. -6.1 + 6.1 + n = 5.36749476 + 6.1 Combine like terms: -6.1 + 6.1 = 0.0 0.0 + n = 5.36749476 + 6.1 n = 5.36749476 + 6.1 Combine like terms: 5.36749476 + 6.1 = 11.46749476 n = 11.46749476 Simplifying n = 11.46749476Subproblem 2
n + -6.1 = -5.36749476 Simplifying n + -6.1 = -5.36749476 Reorder the terms: -6.1 + n = -5.36749476 Solving -6.1 + n = -5.36749476 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.1' to each side of the equation. -6.1 + 6.1 + n = -5.36749476 + 6.1 Combine like terms: -6.1 + 6.1 = 0.0 0.0 + n = -5.36749476 + 6.1 n = -5.36749476 + 6.1 Combine like terms: -5.36749476 + 6.1 = 0.73250524 n = 0.73250524 Simplifying n = 0.73250524Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.46749476, 0.73250524}Solution
n = {11.46749476, 0.73250524}
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