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Simplifying 4n2 + 6n + -16 = -5n2 Reorder the terms: -16 + 6n + 4n2 = -5n2 Solving -16 + 6n + 4n2 = -5n2 Solving for variable 'n'. Combine like terms: 4n2 + 5n2 = 9n2 -16 + 6n + 9n2 = -5n2 + 5n2 Combine like terms: -5n2 + 5n2 = 0 -16 + 6n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + 0.6666666667n + n2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 0.6666666667n + 1.777777778 + n2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 0.6666666667n + n2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 0.6666666667n + n2 = 0 + 1.777777778 0.6666666667n + n2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 0.6666666667n + n2 = 1.777777778 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 = 1.777777778 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667n + n2 = 1.777777778 + 0.1111111112 Combine like terms: 1.777777778 + 0.1111111112 = 1.8888888892 0.1111111112 + 0.6666666667n + n2 = 1.8888888892 Factor a perfect square on the left side: (n + 0.3333333334)(n + 0.3333333334) = 1.8888888892 Calculate the square root of the right side: 1.374368542 Break this problem into two subproblems by setting (n + 0.3333333334) equal to 1.374368542 and -1.374368542.Subproblem 1
n + 0.3333333334 = 1.374368542 Simplifying n + 0.3333333334 = 1.374368542 Reorder the terms: 0.3333333334 + n = 1.374368542 Solving 0.3333333334 + n = 1.374368542 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 = 1.374368542 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 1.374368542 + -0.3333333334 n = 1.374368542 + -0.3333333334 Combine like terms: 1.374368542 + -0.3333333334 = 1.0410352086 n = 1.0410352086 Simplifying n = 1.0410352086Subproblem 2
n + 0.3333333334 = -1.374368542 Simplifying n + 0.3333333334 = -1.374368542 Reorder the terms: 0.3333333334 + n = -1.374368542 Solving 0.3333333334 + n = -1.374368542 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 = -1.374368542 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -1.374368542 + -0.3333333334 n = -1.374368542 + -0.3333333334 Combine like terms: -1.374368542 + -0.3333333334 = -1.7077018754 n = -1.7077018754 Simplifying n = -1.7077018754Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.0410352086, -1.7077018754}
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