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Simplifying 3n2 + -10n + -15 = 0 Reorder the terms: -15 + -10n + 3n2 = 0 Solving -15 + -10n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5 + -3.333333333n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -3.333333333n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -3.333333333n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -3.333333333n + n2 = 0 + 5 -3.333333333n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -3.333333333n + n2 = 5 The n term is -3.333333333n. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333n + 2.777777779 + n2 = 5 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333n + n2 = 5 + 2.777777779 Combine like terms: 5 + 2.777777779 = 7.777777779 2.777777779 + -3.333333333n + n2 = 7.777777779 Factor a perfect square on the left side: (n + -1.666666667)(n + -1.666666667) = 7.777777779 Calculate the square root of the right side: 2.788866755 Break this problem into two subproblems by setting (n + -1.666666667) equal to 2.788866755 and -2.788866755.Subproblem 1
n + -1.666666667 = 2.788866755 Simplifying n + -1.666666667 = 2.788866755 Reorder the terms: -1.666666667 + n = 2.788866755 Solving -1.666666667 + n = 2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + n = 2.788866755 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + n = 2.788866755 + 1.666666667 n = 2.788866755 + 1.666666667 Combine like terms: 2.788866755 + 1.666666667 = 4.455533422 n = 4.455533422 Simplifying n = 4.455533422Subproblem 2
n + -1.666666667 = -2.788866755 Simplifying n + -1.666666667 = -2.788866755 Reorder the terms: -1.666666667 + n = -2.788866755 Solving -1.666666667 + n = -2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + n = -2.788866755 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + n = -2.788866755 + 1.666666667 n = -2.788866755 + 1.666666667 Combine like terms: -2.788866755 + 1.666666667 = -1.122200088 n = -1.122200088 Simplifying n = -1.122200088Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.455533422, -1.122200088}
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