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Simplifying 9n + -3n2 = 4.75 Solving 9n + -3n2 = 4.75 Solving for variable 'n'. Reorder the terms: -4.75 + 9n + -3n2 = 4.75 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 -4.75 + 9n + -3n2 = 0.00 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 1.583333333 + -3n + n2 = 0.00 Move the constant term to the right: Add '-1.583333333' to each side of the equation. 1.583333333 + -3n + -1.583333333 + n2 = 0.00 + -1.583333333 Reorder the terms: 1.583333333 + -1.583333333 + -3n + n2 = 0.00 + -1.583333333 Combine like terms: 1.583333333 + -1.583333333 = 0.000000000 0.000000000 + -3n + n2 = 0.00 + -1.583333333 -3n + n2 = 0.00 + -1.583333333 Combine like terms: 0.00 + -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 = 0.666666667 2.25 + -3n + n2 = 0.666666667 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 0.666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (n + -1.5) equal to 0.816496581 and -0.816496581.Subproblem 1
n + -1.5 = 0.816496581 Simplifying n + -1.5 = 0.816496581 Reorder the terms: -1.5 + n = 0.816496581 Solving -1.5 + n = 0.816496581 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 = 0.816496581 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 0.816496581 + 1.5 n = 0.816496581 + 1.5 Combine like terms: 0.816496581 + 1.5 = 2.316496581 n = 2.316496581 Simplifying n = 2.316496581Subproblem 2
n + -1.5 = -0.816496581 Simplifying n + -1.5 = -0.816496581 Reorder the terms: -1.5 + n = -0.816496581 Solving -1.5 + n = -0.816496581 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 = -0.816496581 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -0.816496581 + 1.5 n = -0.816496581 + 1.5 Combine like terms: -0.816496581 + 1.5 = 0.683503419 n = 0.683503419 Simplifying n = 0.683503419Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.316496581, 0.683503419}
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