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Simplifying 3n2 + -13 = 9n Reorder the terms: -13 + 3n2 = 9n Solving -13 + 3n2 = 9n Solving for variable 'n'. Reorder the terms: -13 + -9n + 3n2 = 9n + -9n Combine like terms: 9n + -9n = 0 -13 + -9n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4.333333333 + -3n + n2 = 0 Move the constant term to the right: Add '4.333333333' to each side of the equation. -4.333333333 + -3n + 4.333333333 + n2 = 0 + 4.333333333 Reorder the terms: -4.333333333 + 4.333333333 + -3n + n2 = 0 + 4.333333333 Combine like terms: -4.333333333 + 4.333333333 = 0.000000000 0.000000000 + -3n + n2 = 0 + 4.333333333 -3n + n2 = 0 + 4.333333333 Combine like terms: 0 + 4.333333333 = 4.333333333 -3n + n2 = 4.333333333 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 4.333333333 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 4.333333333 + 2.25 Combine like terms: 4.333333333 + 2.25 = 6.583333333 2.25 + -3n + n2 = 6.583333333 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 6.583333333 Calculate the square root of the right side: 2.56580072 Break this problem into two subproblems by setting (n + -1.5) equal to 2.56580072 and -2.56580072.Subproblem 1
n + -1.5 = 2.56580072 Simplifying n + -1.5 = 2.56580072 Reorder the terms: -1.5 + n = 2.56580072 Solving -1.5 + n = 2.56580072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 2.56580072 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.56580072 + 1.5 n = 2.56580072 + 1.5 Combine like terms: 2.56580072 + 1.5 = 4.06580072 n = 4.06580072 Simplifying n = 4.06580072Subproblem 2
n + -1.5 = -2.56580072 Simplifying n + -1.5 = -2.56580072 Reorder the terms: -1.5 + n = -2.56580072 Solving -1.5 + n = -2.56580072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -2.56580072 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.56580072 + 1.5 n = -2.56580072 + 1.5 Combine like terms: -2.56580072 + 1.5 = -1.06580072 n = -1.06580072 Simplifying n = -1.06580072Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.06580072, -1.06580072}
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