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Simplifying 7n2 + -2n + -14 = 21 Reorder the terms: -14 + -2n + 7n2 = 21 Solving -14 + -2n + 7n2 = 21 Solving for variable 'n'. Reorder the terms: -14 + -21 + -2n + 7n2 = 21 + -21 Combine like terms: -14 + -21 = -35 -35 + -2n + 7n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -35 + -2n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5 + -0.2857142857n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.2857142857n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.2857142857n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.2857142857n + n2 = 0 + 5 -0.2857142857n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.2857142857n + n2 = 5 The n term is -0.2857142857n. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857n + 0.02040816328 + n2 = 5 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857n + n2 = 5 + 0.02040816328 Combine like terms: 5 + 0.02040816328 = 5.02040816328 0.02040816328 + -0.2857142857n + n2 = 5.02040816328 Factor a perfect square on the left side: (n + -0.1428571429)(n + -0.1428571429) = 5.02040816328 Calculate the square root of the right side: 2.240626734 Break this problem into two subproblems by setting (n + -0.1428571429) equal to 2.240626734 and -2.240626734.Subproblem 1
n + -0.1428571429 = 2.240626734 Simplifying n + -0.1428571429 = 2.240626734 Reorder the terms: -0.1428571429 + n = 2.240626734 Solving -0.1428571429 + n = 2.240626734 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + n = 2.240626734 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = 2.240626734 + 0.1428571429 n = 2.240626734 + 0.1428571429 Combine like terms: 2.240626734 + 0.1428571429 = 2.3834838769 n = 2.3834838769 Simplifying n = 2.3834838769Subproblem 2
n + -0.1428571429 = -2.240626734 Simplifying n + -0.1428571429 = -2.240626734 Reorder the terms: -0.1428571429 + n = -2.240626734 Solving -0.1428571429 + n = -2.240626734 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + n = -2.240626734 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = -2.240626734 + 0.1428571429 n = -2.240626734 + 0.1428571429 Combine like terms: -2.240626734 + 0.1428571429 = -2.0977695911 n = -2.0977695911 Simplifying n = -2.0977695911Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.3834838769, -2.0977695911}
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