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Simplifying -3n(5 + 7n) = -120 (5 * -3n + 7n * -3n) = -120 (-15n + -21n2) = -120 Solving -15n + -21n2 = -120 Solving for variable 'n'. Reorder the terms: 120 + -15n + -21n2 = -120 + 120 Combine like terms: -120 + 120 = 0 120 + -15n + -21n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(40 + -5n + -7n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(40 + -5n + -7n2)' equal to zero and attempt to solve: Simplifying 40 + -5n + -7n2 = 0 Solving 40 + -5n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -5.714285714 + 0.7142857143n + n2 = 0 Move the constant term to the right: Add '5.714285714' to each side of the equation. -5.714285714 + 0.7142857143n + 5.714285714 + n2 = 0 + 5.714285714 Reorder the terms: -5.714285714 + 5.714285714 + 0.7142857143n + n2 = 0 + 5.714285714 Combine like terms: -5.714285714 + 5.714285714 = 0.000000000 0.000000000 + 0.7142857143n + n2 = 0 + 5.714285714 0.7142857143n + n2 = 0 + 5.714285714 Combine like terms: 0 + 5.714285714 = 5.714285714 0.7142857143n + n2 = 5.714285714 The n term is 0.7142857143n. Take half its coefficient (0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. 0.7142857143n + 0.1275510204 + n2 = 5.714285714 + 0.1275510204 Reorder the terms: 0.1275510204 + 0.7142857143n + n2 = 5.714285714 + 0.1275510204 Combine like terms: 5.714285714 + 0.1275510204 = 5.8418367344 0.1275510204 + 0.7142857143n + n2 = 5.8418367344 Factor a perfect square on the left side: (n + 0.3571428572)(n + 0.3571428572) = 5.8418367344 Calculate the square root of the right side: 2.416989188 Break this problem into two subproblems by setting (n + 0.3571428572) equal to 2.416989188 and -2.416989188.Subproblem 1
n + 0.3571428572 = 2.416989188 Simplifying n + 0.3571428572 = 2.416989188 Reorder the terms: 0.3571428572 + n = 2.416989188 Solving 0.3571428572 + n = 2.416989188 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3571428572' to each side of the equation. 0.3571428572 + -0.3571428572 + n = 2.416989188 + -0.3571428572 Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000 0.0000000000 + n = 2.416989188 + -0.3571428572 n = 2.416989188 + -0.3571428572 Combine like terms: 2.416989188 + -0.3571428572 = 2.0598463308 n = 2.0598463308 Simplifying n = 2.0598463308Subproblem 2
n + 0.3571428572 = -2.416989188 Simplifying n + 0.3571428572 = -2.416989188 Reorder the terms: 0.3571428572 + n = -2.416989188 Solving 0.3571428572 + n = -2.416989188 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3571428572' to each side of the equation. 0.3571428572 + -0.3571428572 + n = -2.416989188 + -0.3571428572 Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000 0.0000000000 + n = -2.416989188 + -0.3571428572 n = -2.416989188 + -0.3571428572 Combine like terms: -2.416989188 + -0.3571428572 = -2.7741320452 n = -2.7741320452 Simplifying n = -2.7741320452Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.0598463308, -2.7741320452}Solution
n = {2.0598463308, -2.7741320452}
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