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Simplifying 9n + 4 = 3n2 + -1(0.75) Reorder the terms: 4 + 9n = 3n2 + -1(0.75) Multiply -1 * 0.75 4 + 9n = 3n2 + -0.75 Reorder the terms: 4 + 9n = -0.75 + 3n2 Solving 4 + 9n = -0.75 + 3n2 Solving for variable 'n'. Reorder the terms: 4 + 0.75 + 9n + -3n2 = -0.75 + 3n2 + 0.75 + -3n2 Combine like terms: 4 + 0.75 = 4.75 4.75 + 9n + -3n2 = -0.75 + 3n2 + 0.75 + -3n2 Reorder the terms: 4.75 + 9n + -3n2 = -0.75 + 0.75 + 3n2 + -3n2 Combine like terms: -0.75 + 0.75 = 0.00 4.75 + 9n + -3n2 = 0.00 + 3n2 + -3n2 4.75 + 9n + -3n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 4.75 + 9n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1.583333333 + -3n + n2 = 0 Move the constant term to the right: Add '1.583333333' to each side of the equation. -1.583333333 + -3n + 1.583333333 + n2 = 0 + 1.583333333 Reorder the terms: -1.583333333 + 1.583333333 + -3n + n2 = 0 + 1.583333333 Combine like terms: -1.583333333 + 1.583333333 = 0.000000000 0.000000000 + -3n + n2 = 0 + 1.583333333 -3n + n2 = 0 + 1.583333333 Combine like terms: 0 + 1.583333333 = 1.583333333 -3n + n2 = 1.583333333 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 1.583333333 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 1.583333333 + 2.25 Combine like terms: 1.583333333 + 2.25 = 3.833333333 2.25 + -3n + n2 = 3.833333333 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 3.833333333 Calculate the square root of the right side: 1.957890021 Break this problem into two subproblems by setting (n + -1.5) equal to 1.957890021 and -1.957890021.Subproblem 1
n + -1.5 = 1.957890021 Simplifying n + -1.5 = 1.957890021 Reorder the terms: -1.5 + n = 1.957890021 Solving -1.5 + n = 1.957890021 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 1.957890021 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.957890021 + 1.5 n = 1.957890021 + 1.5 Combine like terms: 1.957890021 + 1.5 = 3.457890021 n = 3.457890021 Simplifying n = 3.457890021Subproblem 2
n + -1.5 = -1.957890021 Simplifying n + -1.5 = -1.957890021 Reorder the terms: -1.5 + n = -1.957890021 Solving -1.5 + n = -1.957890021 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -1.957890021 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.957890021 + 1.5 n = -1.957890021 + 1.5 Combine like terms: -1.957890021 + 1.5 = -0.457890021 n = -0.457890021 Simplifying n = -0.457890021Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.457890021, -0.457890021}
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