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Simplifying -0.25n2 + 6n + -27 = 5 Reorder the terms: -27 + 6n + -0.25n2 = 5 Solving -27 + 6n + -0.25n2 = 5 Solving for variable 'n'. Reorder the terms: -27 + -5 + 6n + -0.25n2 = 5 + -5 Combine like terms: -27 + -5 = -32 -32 + 6n + -0.25n2 = 5 + -5 Combine like terms: 5 + -5 = 0 -32 + 6n + -0.25n2 = 0 Begin completing the square. Divide all terms by -0.25 the coefficient of the squared term: Divide each side by '-0.25'. 128 + -24n + n2 = 0 Move the constant term to the right: Add '-128' to each side of the equation. 128 + -24n + -128 + n2 = 0 + -128 Reorder the terms: 128 + -128 + -24n + n2 = 0 + -128 Combine like terms: 128 + -128 = 0 0 + -24n + n2 = 0 + -128 -24n + n2 = 0 + -128 Combine like terms: 0 + -128 = -128 -24n + n2 = -128 The n term is -24n. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24n + 144 + n2 = -128 + 144 Reorder the terms: 144 + -24n + n2 = -128 + 144 Combine like terms: -128 + 144 = 16 144 + -24n + n2 = 16 Factor a perfect square on the left side: (n + -12)(n + -12) = 16 Calculate the square root of the right side: 4 Break this problem into two subproblems by setting (n + -12) equal to 4 and -4.Subproblem 1
n + -12 = 4 Simplifying n + -12 = 4 Reorder the terms: -12 + n = 4 Solving -12 + n = 4 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + n = 4 + 12 Combine like terms: -12 + 12 = 0 0 + n = 4 + 12 n = 4 + 12 Combine like terms: 4 + 12 = 16 n = 16 Simplifying n = 16Subproblem 2
n + -12 = -4 Simplifying n + -12 = -4 Reorder the terms: -12 + n = -4 Solving -12 + n = -4 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + n = -4 + 12 Combine like terms: -12 + 12 = 0 0 + n = -4 + 12 n = -4 + 12 Combine like terms: -4 + 12 = 8 n = 8 Simplifying n = 8Solution
The solution to the problem is based on the solutions from the subproblems. n = {16, 8}
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