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Simplifying 9n + -4 = 3n2 + -0.75 Reorder the terms: -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 = -3.25 -3.25 + 9n + -3n2 = -0.75 + 3n2 + 0.75 + -3n2 Reorder the terms: -3.25 + 9n + -3n2 = -0.75 + 0.75 + 3n2 + -3n2 Combine like terms: -0.75 + 0.75 = 0.00 -3.25 + 9n + -3n2 = 0.00 + 3n2 + -3n2 -3.25 + 9n + -3n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 -3.25 + 9n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 1.083333333 + -3n + n2 = 0 Move the constant term to the right: Add '-1.083333333' to each side of the equation. 1.083333333 + -3n + -1.083333333 + n2 = 0 + -1.083333333 Reorder the terms: 1.083333333 + -1.083333333 + -3n + n2 = 0 + -1.083333333 Combine like terms: 1.083333333 + -1.083333333 = 0.000000000 0.000000000 + -3n + n2 = 0 + -1.083333333 -3n + n2 = 0 + -1.083333333 Combine like terms: 0 + -1.083333333 = -1.083333333 -3n + n2 = -1.083333333 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 = -1.083333333 + 2.25 Reorder the terms: 2.25 + -3n + n2 = -1.083333333 + 2.25 Combine like terms: -1.083333333 + 2.25 = 1.166666667 2.25 + -3n + n2 = 1.166666667 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 1.166666667 Calculate the square root of the right side: 1.08012345 Break this problem into two subproblems by setting (n + -1.5) equal to 1.08012345 and -1.08012345.Subproblem 1
n + -1.5 = 1.08012345 Simplifying n + -1.5 = 1.08012345 Reorder the terms: -1.5 + n = 1.08012345 Solving -1.5 + n = 1.08012345 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 = 1.08012345 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.08012345 + 1.5 n = 1.08012345 + 1.5 Combine like terms: 1.08012345 + 1.5 = 2.58012345 n = 2.58012345 Simplifying n = 2.58012345Subproblem 2
n + -1.5 = -1.08012345 Simplifying n + -1.5 = -1.08012345 Reorder the terms: -1.5 + n = -1.08012345 Solving -1.5 + n = -1.08012345 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 = -1.08012345 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.08012345 + 1.5 n = -1.08012345 + 1.5 Combine like terms: -1.08012345 + 1.5 = 0.41987655 n = 0.41987655 Simplifying n = 0.41987655Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.58012345, 0.41987655}
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