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Simplifying 6n2 + -114n = -14 Reorder the terms: -114n + 6n2 = -14 Solving -114n + 6n2 = -14 Solving for variable 'n'. Reorder the terms: 14 + -114n + 6n2 = -14 + 14 Combine like terms: -14 + 14 = 0 14 + -114n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -57n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -57n + 3n2)' equal to zero and attempt to solve: Simplifying 7 + -57n + 3n2 = 0 Solving 7 + -57n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2.333333333 + -19n + n2 = 0 Move the constant term to the right: Add '-2.333333333' to each side of the equation. 2.333333333 + -19n + -2.333333333 + n2 = 0 + -2.333333333 Reorder the terms: 2.333333333 + -2.333333333 + -19n + n2 = 0 + -2.333333333 Combine like terms: 2.333333333 + -2.333333333 = 0.000000000 0.000000000 + -19n + n2 = 0 + -2.333333333 -19n + n2 = 0 + -2.333333333 Combine like terms: 0 + -2.333333333 = -2.333333333 -19n + n2 = -2.333333333 The n term is -19n. Take half its coefficient (-9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. -19n + 90.25 + n2 = -2.333333333 + 90.25 Reorder the terms: 90.25 + -19n + n2 = -2.333333333 + 90.25 Combine like terms: -2.333333333 + 90.25 = 87.916666667 90.25 + -19n + n2 = 87.916666667 Factor a perfect square on the left side: (n + -9.5)(n + -9.5) = 87.916666667 Calculate the square root of the right side: 9.376388786 Break this problem into two subproblems by setting (n + -9.5) equal to 9.376388786 and -9.376388786.Subproblem 1
n + -9.5 = 9.376388786 Simplifying n + -9.5 = 9.376388786 Reorder the terms: -9.5 + n = 9.376388786 Solving -9.5 + n = 9.376388786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + n = 9.376388786 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + n = 9.376388786 + 9.5 n = 9.376388786 + 9.5 Combine like terms: 9.376388786 + 9.5 = 18.876388786 n = 18.876388786 Simplifying n = 18.876388786Subproblem 2
n + -9.5 = -9.376388786 Simplifying n + -9.5 = -9.376388786 Reorder the terms: -9.5 + n = -9.376388786 Solving -9.5 + n = -9.376388786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + n = -9.376388786 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + n = -9.376388786 + 9.5 n = -9.376388786 + 9.5 Combine like terms: -9.376388786 + 9.5 = 0.123611214 n = 0.123611214 Simplifying n = 0.123611214Solution
The solution to the problem is based on the solutions from the subproblems. n = {18.876388786, 0.123611214}Solution
n = {18.876388786, 0.123611214}
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