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Simplifying 6n2 + 10n + -1010 = 0 Reorder the terms: -1010 + 10n + 6n2 = 0 Solving -1010 + 10n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-505 + 5n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-505 + 5n + 3n2)' equal to zero and attempt to solve: Simplifying -505 + 5n + 3n2 = 0 Solving -505 + 5n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -168.3333333 + 1.666666667n + n2 = 0 Move the constant term to the right: Add '168.3333333' to each side of the equation. -168.3333333 + 1.666666667n + 168.3333333 + n2 = 0 + 168.3333333 Reorder the terms: -168.3333333 + 168.3333333 + 1.666666667n + n2 = 0 + 168.3333333 Combine like terms: -168.3333333 + 168.3333333 = 0.0000000 0.0000000 + 1.666666667n + n2 = 0 + 168.3333333 1.666666667n + n2 = 0 + 168.3333333 Combine like terms: 0 + 168.3333333 = 168.3333333 1.666666667n + n2 = 168.3333333 The n term is 1.666666667n. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667n + 0.6944444447 + n2 = 168.3333333 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667n + n2 = 168.3333333 + 0.6944444447 Combine like terms: 168.3333333 + 0.6944444447 = 169.0277777447 0.6944444447 + 1.666666667n + n2 = 169.0277777447 Factor a perfect square on the left side: (n + 0.8333333335)(n + 0.8333333335) = 169.0277777447 Calculate the square root of the right side: 13.001068331 Break this problem into two subproblems by setting (n + 0.8333333335) equal to 13.001068331 and -13.001068331.Subproblem 1
n + 0.8333333335 = 13.001068331 Simplifying n + 0.8333333335 = 13.001068331 Reorder the terms: 0.8333333335 + n = 13.001068331 Solving 0.8333333335 + n = 13.001068331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + n = 13.001068331 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + n = 13.001068331 + -0.8333333335 n = 13.001068331 + -0.8333333335 Combine like terms: 13.001068331 + -0.8333333335 = 12.1677349975 n = 12.1677349975 Simplifying n = 12.1677349975Subproblem 2
n + 0.8333333335 = -13.001068331 Simplifying n + 0.8333333335 = -13.001068331 Reorder the terms: 0.8333333335 + n = -13.001068331 Solving 0.8333333335 + n = -13.001068331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + n = -13.001068331 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + n = -13.001068331 + -0.8333333335 n = -13.001068331 + -0.8333333335 Combine like terms: -13.001068331 + -0.8333333335 = -13.8344016645 n = -13.8344016645 Simplifying n = -13.8344016645Solution
The solution to the problem is based on the solutions from the subproblems. n = {12.1677349975, -13.8344016645}Solution
n = {12.1677349975, -13.8344016645}
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