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Simplifying 3n2 = 19 + -1n Solving 3n2 = 19 + -1n Solving for variable 'n'. Reorder the terms: -19 + n + 3n2 = 19 + -1n + -19 + n Reorder the terms: -19 + n + 3n2 = 19 + -19 + -1n + n Combine like terms: 19 + -19 = 0 -19 + n + 3n2 = 0 + -1n + n -19 + n + 3n2 = -1n + n Combine like terms: -1n + n = 0 -19 + n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6.333333333 + 0.3333333333n + n2 = 0 Move the constant term to the right: Add '6.333333333' to each side of the equation. -6.333333333 + 0.3333333333n + 6.333333333 + n2 = 0 + 6.333333333 Reorder the terms: -6.333333333 + 6.333333333 + 0.3333333333n + n2 = 0 + 6.333333333 Combine like terms: -6.333333333 + 6.333333333 = 0.000000000 0.000000000 + 0.3333333333n + n2 = 0 + 6.333333333 0.3333333333n + n2 = 0 + 6.333333333 Combine like terms: 0 + 6.333333333 = 6.333333333 0.3333333333n + n2 = 6.333333333 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333n + 0.25 + n2 = 6.333333333 + 0.25 Reorder the terms: 0.25 + 0.3333333333n + n2 = 6.333333333 + 0.25 Combine like terms: 6.333333333 + 0.25 = 6.583333333 0.25 + 0.3333333333n + n2 = 6.583333333 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 6.583333333 Calculate the square root of the right side: 2.56580072 Break this problem into two subproblems by setting (n + 0.5) equal to 2.56580072 and -2.56580072.Subproblem 1
n + 0.5 = 2.56580072 Simplifying n + 0.5 = 2.56580072 Reorder the terms: 0.5 + n = 2.56580072 Solving 0.5 + n = 2.56580072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 2.56580072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 2.56580072 + -0.5 n = 2.56580072 + -0.5 Combine like terms: 2.56580072 + -0.5 = 2.06580072 n = 2.06580072 Simplifying n = 2.06580072Subproblem 2
n + 0.5 = -2.56580072 Simplifying n + 0.5 = -2.56580072 Reorder the terms: 0.5 + n = -2.56580072 Solving 0.5 + n = -2.56580072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -2.56580072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -2.56580072 + -0.5 n = -2.56580072 + -0.5 Combine like terms: -2.56580072 + -0.5 = -3.06580072 n = -3.06580072 Simplifying n = -3.06580072Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.06580072, -3.06580072}
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