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Simplifying 4n2 + -93 = 5n Reorder the terms: -93 + 4n2 = 5n Solving -93 + 4n2 = 5n Solving for variable 'n'. Reorder the terms: -93 + -5n + 4n2 = 5n + -5n Combine like terms: 5n + -5n = 0 -93 + -5n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -23.25 + -1.25n + n2 = 0 Move the constant term to the right: Add '23.25' to each side of the equation. -23.25 + -1.25n + 23.25 + n2 = 0 + 23.25 Reorder the terms: -23.25 + 23.25 + -1.25n + n2 = 0 + 23.25 Combine like terms: -23.25 + 23.25 = 0.00 0.00 + -1.25n + n2 = 0 + 23.25 -1.25n + n2 = 0 + 23.25 Combine like terms: 0 + 23.25 = 23.25 -1.25n + n2 = 23.25 The n term is -1.25n. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25n + 0.390625 + n2 = 23.25 + 0.390625 Reorder the terms: 0.390625 + -1.25n + n2 = 23.25 + 0.390625 Combine like terms: 23.25 + 0.390625 = 23.640625 0.390625 + -1.25n + n2 = 23.640625 Factor a perfect square on the left side: (n + -0.625)(n + -0.625) = 23.640625 Calculate the square root of the right side: 4.862162585 Break this problem into two subproblems by setting (n + -0.625) equal to 4.862162585 and -4.862162585.Subproblem 1
n + -0.625 = 4.862162585 Simplifying n + -0.625 = 4.862162585 Reorder the terms: -0.625 + n = 4.862162585 Solving -0.625 + n = 4.862162585 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = 4.862162585 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = 4.862162585 + 0.625 n = 4.862162585 + 0.625 Combine like terms: 4.862162585 + 0.625 = 5.487162585 n = 5.487162585 Simplifying n = 5.487162585Subproblem 2
n + -0.625 = -4.862162585 Simplifying n + -0.625 = -4.862162585 Reorder the terms: -0.625 + n = -4.862162585 Solving -0.625 + n = -4.862162585 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = -4.862162585 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = -4.862162585 + 0.625 n = -4.862162585 + 0.625 Combine like terms: -4.862162585 + 0.625 = -4.237162585 n = -4.237162585 Simplifying n = -4.237162585Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.487162585, -4.237162585}
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