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Simplifying 21n2 + 8 + -4 = 6n + 11 Reorder the terms: 8 + -4 + 21n2 = 6n + 11 Combine like terms: 8 + -4 = 4 4 + 21n2 = 6n + 11 Reorder the terms: 4 + 21n2 = 11 + 6n Solving 4 + 21n2 = 11 + 6n Solving for variable 'n'. Reorder the terms: 4 + -11 + -6n + 21n2 = 11 + 6n + -11 + -6n Combine like terms: 4 + -11 = -7 -7 + -6n + 21n2 = 11 + 6n + -11 + -6n Reorder the terms: -7 + -6n + 21n2 = 11 + -11 + 6n + -6n Combine like terms: 11 + -11 = 0 -7 + -6n + 21n2 = 0 + 6n + -6n -7 + -6n + 21n2 = 6n + -6n Combine like terms: 6n + -6n = 0 -7 + -6n + 21n2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.3333333333 + -0.2857142857n + n2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.2857142857n + 0.3333333333 + n2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.2857142857n + n2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.2857142857n + n2 = 0 + 0.3333333333 -0.2857142857n + n2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.2857142857n + n2 = 0.3333333333 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 = 0.3333333333 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857n + n2 = 0.3333333333 + 0.02040816328 Combine like terms: 0.3333333333 + 0.02040816328 = 0.35374149658 0.02040816328 + -0.2857142857n + n2 = 0.35374149658 Factor a perfect square on the left side: (n + -0.1428571429)(n + -0.1428571429) = 0.35374149658 Calculate the square root of the right side: 0.594761714 Break this problem into two subproblems by setting (n + -0.1428571429) equal to 0.594761714 and -0.594761714.Subproblem 1
n + -0.1428571429 = 0.594761714 Simplifying n + -0.1428571429 = 0.594761714 Reorder the terms: -0.1428571429 + n = 0.594761714 Solving -0.1428571429 + n = 0.594761714 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 = 0.594761714 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = 0.594761714 + 0.1428571429 n = 0.594761714 + 0.1428571429 Combine like terms: 0.594761714 + 0.1428571429 = 0.7376188569 n = 0.7376188569 Simplifying n = 0.7376188569Subproblem 2
n + -0.1428571429 = -0.594761714 Simplifying n + -0.1428571429 = -0.594761714 Reorder the terms: -0.1428571429 + n = -0.594761714 Solving -0.1428571429 + n = -0.594761714 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 = -0.594761714 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = -0.594761714 + 0.1428571429 n = -0.594761714 + 0.1428571429 Combine like terms: -0.594761714 + 0.1428571429 = -0.4519045711 n = -0.4519045711 Simplifying n = -0.4519045711Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.7376188569, -0.4519045711}
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