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Simplifying 2n2 + -6n + 3 = 0 Reorder the terms: 3 + -6n + 2n2 = 0 Solving 3 + -6n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -3n + n2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3n + -1.5 + n2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3n + n2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3n + n2 = 0 + -1.5 -3n + n2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3n + n2 = -1.5 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 = -1.5 + 2.25 Reorder the terms: 2.25 + -3n + n2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3n + n2 = 0.75 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (n + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
n + -1.5 = 0.866025404 Simplifying n + -1.5 = 0.866025404 Reorder the terms: -1.5 + n = 0.866025404 Solving -1.5 + n = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 0.866025404 + 1.5 n = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 n = 2.366025404 Simplifying n = 2.366025404Subproblem 2
n + -1.5 = -0.866025404 Simplifying n + -1.5 = -0.866025404 Reorder the terms: -1.5 + n = -0.866025404 Solving -1.5 + n = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -0.866025404 + 1.5 n = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 n = 0.633974596 Simplifying n = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.366025404, 0.633974596}
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