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Simplifying 2 = -15n(-3n + 2) Reorder the terms: 2 = -15n(2 + -3n) 2 = (2 * -15n + -3n * -15n) 2 = (-30n + 45n2) Solving 2 = -30n + 45n2 Solving for variable 'n'. Reorder the terms: 2 + 30n + -45n2 = -30n + 30n + 45n2 + -45n2 Combine like terms: -30n + 30n = 0 2 + 30n + -45n2 = 0 + 45n2 + -45n2 2 + 30n + -45n2 = 45n2 + -45n2 Combine like terms: 45n2 + -45n2 = 0 2 + 30n + -45n2 = 0 Begin completing the square. Divide all terms by -45 the coefficient of the squared term: Divide each side by '-45'. -0.04444444444 + -0.6666666667n + n2 = 0 Move the constant term to the right: Add '0.04444444444' to each side of the equation. -0.04444444444 + -0.6666666667n + 0.04444444444 + n2 = 0 + 0.04444444444 Reorder the terms: -0.04444444444 + 0.04444444444 + -0.6666666667n + n2 = 0 + 0.04444444444 Combine like terms: -0.04444444444 + 0.04444444444 = 0.00000000000 0.00000000000 + -0.6666666667n + n2 = 0 + 0.04444444444 -0.6666666667n + n2 = 0 + 0.04444444444 Combine like terms: 0 + 0.04444444444 = 0.04444444444 -0.6666666667n + n2 = 0.04444444444 The n term is -0.6666666667n. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667n + 0.1111111112 + n2 = 0.04444444444 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667n + n2 = 0.04444444444 + 0.1111111112 Combine like terms: 0.04444444444 + 0.1111111112 = 0.15555555564 0.1111111112 + -0.6666666667n + n2 = 0.15555555564 Factor a perfect square on the left side: (n + -0.3333333334)(n + -0.3333333334) = 0.15555555564 Calculate the square root of the right side: 0.394405319 Break this problem into two subproblems by setting (n + -0.3333333334) equal to 0.394405319 and -0.394405319.Subproblem 1
n + -0.3333333334 = 0.394405319 Simplifying n + -0.3333333334 = 0.394405319 Reorder the terms: -0.3333333334 + n = 0.394405319 Solving -0.3333333334 + n = 0.394405319 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = 0.394405319 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 0.394405319 + 0.3333333334 n = 0.394405319 + 0.3333333334 Combine like terms: 0.394405319 + 0.3333333334 = 0.7277386524 n = 0.7277386524 Simplifying n = 0.7277386524Subproblem 2
n + -0.3333333334 = -0.394405319 Simplifying n + -0.3333333334 = -0.394405319 Reorder the terms: -0.3333333334 + n = -0.394405319 Solving -0.3333333334 + n = -0.394405319 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = -0.394405319 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -0.394405319 + 0.3333333334 n = -0.394405319 + 0.3333333334 Combine like terms: -0.394405319 + 0.3333333334 = -0.0610719856 n = -0.0610719856 Simplifying n = -0.0610719856Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.7277386524, -0.0610719856}
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