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Simplifying 3n2 + -15 = -2n Reorder the terms: -15 + 3n2 = -2n Solving -15 + 3n2 = -2n Solving for variable 'n'. Reorder the terms: -15 + 2n + 3n2 = -2n + 2n Combine like terms: -2n + 2n = 0 -15 + 2n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5 + 0.6666666667n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 0.6666666667n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + 0.6666666667n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 0.6666666667n + n2 = 0 + 5 0.6666666667n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 0.6666666667n + n2 = 5 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 = 5 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667n + n2 = 5 + 0.1111111112 Combine like terms: 5 + 0.1111111112 = 5.1111111112 0.1111111112 + 0.6666666667n + n2 = 5.1111111112 Factor a perfect square on the left side: (n + 0.3333333334)(n + 0.3333333334) = 5.1111111112 Calculate the square root of the right side: 2.260776661 Break this problem into two subproblems by setting (n + 0.3333333334) equal to 2.260776661 and -2.260776661.Subproblem 1
n + 0.3333333334 = 2.260776661 Simplifying n + 0.3333333334 = 2.260776661 Reorder the terms: 0.3333333334 + n = 2.260776661 Solving 0.3333333334 + n = 2.260776661 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 = 2.260776661 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 2.260776661 + -0.3333333334 n = 2.260776661 + -0.3333333334 Combine like terms: 2.260776661 + -0.3333333334 = 1.9274433276 n = 1.9274433276 Simplifying n = 1.9274433276Subproblem 2
n + 0.3333333334 = -2.260776661 Simplifying n + 0.3333333334 = -2.260776661 Reorder the terms: 0.3333333334 + n = -2.260776661 Solving 0.3333333334 + n = -2.260776661 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 = -2.260776661 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -2.260776661 + -0.3333333334 n = -2.260776661 + -0.3333333334 Combine like terms: -2.260776661 + -0.3333333334 = -2.5941099944 n = -2.5941099944 Simplifying n = -2.5941099944Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.9274433276, -2.5941099944}
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