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Simplifying 5n2 + 32n = -21 Reorder the terms: 32n + 5n2 = -21 Solving 32n + 5n2 = -21 Solving for variable 'n'. Reorder the terms: 21 + 32n + 5n2 = -21 + 21 Combine like terms: -21 + 21 = 0 21 + 32n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 4.2 + 6.4n + n2 = 0 Move the constant term to the right: Add '-4.2' to each side of the equation. 4.2 + 6.4n + -4.2 + n2 = 0 + -4.2 Reorder the terms: 4.2 + -4.2 + 6.4n + n2 = 0 + -4.2 Combine like terms: 4.2 + -4.2 = 0.0 0.0 + 6.4n + n2 = 0 + -4.2 6.4n + n2 = 0 + -4.2 Combine like terms: 0 + -4.2 = -4.2 6.4n + n2 = -4.2 The n term is 6.4n. Take half its coefficient (3.2). Square it (10.24) and add it to both sides. Add '10.24' to each side of the equation. 6.4n + 10.24 + n2 = -4.2 + 10.24 Reorder the terms: 10.24 + 6.4n + n2 = -4.2 + 10.24 Combine like terms: -4.2 + 10.24 = 6.04 10.24 + 6.4n + n2 = 6.04 Factor a perfect square on the left side: (n + 3.2)(n + 3.2) = 6.04 Calculate the square root of the right side: 2.457641145 Break this problem into two subproblems by setting (n + 3.2) equal to 2.457641145 and -2.457641145.Subproblem 1
n + 3.2 = 2.457641145 Simplifying n + 3.2 = 2.457641145 Reorder the terms: 3.2 + n = 2.457641145 Solving 3.2 + n = 2.457641145 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.2' to each side of the equation. 3.2 + -3.2 + n = 2.457641145 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + n = 2.457641145 + -3.2 n = 2.457641145 + -3.2 Combine like terms: 2.457641145 + -3.2 = -0.742358855 n = -0.742358855 Simplifying n = -0.742358855Subproblem 2
n + 3.2 = -2.457641145 Simplifying n + 3.2 = -2.457641145 Reorder the terms: 3.2 + n = -2.457641145 Solving 3.2 + n = -2.457641145 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.2' to each side of the equation. 3.2 + -3.2 + n = -2.457641145 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + n = -2.457641145 + -3.2 n = -2.457641145 + -3.2 Combine like terms: -2.457641145 + -3.2 = -5.657641145 n = -5.657641145 Simplifying n = -5.657641145Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.742358855, -5.657641145}
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