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Simplifying 18n + 8 = 3n2 + -0.75 Reorder the terms: 8 + 18n = 3n2 + -0.75 Reorder the terms: 8 + 18n = -0.75 + 3n2 Solving 8 + 18n = -0.75 + 3n2 Solving for variable 'n'. Reorder the terms: 8 + 0.75 + 18n + -3n2 = -0.75 + 3n2 + 0.75 + -3n2 Combine like terms: 8 + 0.75 = 8.75 8.75 + 18n + -3n2 = -0.75 + 3n2 + 0.75 + -3n2 Reorder the terms: 8.75 + 18n + -3n2 = -0.75 + 0.75 + 3n2 + -3n2 Combine like terms: -0.75 + 0.75 = 0.00 8.75 + 18n + -3n2 = 0.00 + 3n2 + -3n2 8.75 + 18n + -3n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 8.75 + 18n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2.916666667 + -6n + n2 = 0 Move the constant term to the right: Add '2.916666667' to each side of the equation. -2.916666667 + -6n + 2.916666667 + n2 = 0 + 2.916666667 Reorder the terms: -2.916666667 + 2.916666667 + -6n + n2 = 0 + 2.916666667 Combine like terms: -2.916666667 + 2.916666667 = 0.000000000 0.000000000 + -6n + n2 = 0 + 2.916666667 -6n + n2 = 0 + 2.916666667 Combine like terms: 0 + 2.916666667 = 2.916666667 -6n + n2 = 2.916666667 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = 2.916666667 + 9 Reorder the terms: 9 + -6n + n2 = 2.916666667 + 9 Combine like terms: 2.916666667 + 9 = 11.916666667 9 + -6n + n2 = 11.916666667 Factor a perfect square on the left side: (n + -3)(n + -3) = 11.916666667 Calculate the square root of the right side: 3.45205253 Break this problem into two subproblems by setting (n + -3) equal to 3.45205253 and -3.45205253.Subproblem 1
n + -3 = 3.45205253 Simplifying n + -3 = 3.45205253 Reorder the terms: -3 + n = 3.45205253 Solving -3 + n = 3.45205253 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 3.45205253 + 3 Combine like terms: -3 + 3 = 0 0 + n = 3.45205253 + 3 n = 3.45205253 + 3 Combine like terms: 3.45205253 + 3 = 6.45205253 n = 6.45205253 Simplifying n = 6.45205253Subproblem 2
n + -3 = -3.45205253 Simplifying n + -3 = -3.45205253 Reorder the terms: -3 + n = -3.45205253 Solving -3 + n = -3.45205253 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -3.45205253 + 3 Combine like terms: -3 + 3 = 0 0 + n = -3.45205253 + 3 n = -3.45205253 + 3 Combine like terms: -3.45205253 + 3 = -0.45205253 n = -0.45205253 Simplifying n = -0.45205253Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.45205253, -0.45205253}
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