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Simplifying 6n2 + -3n = 14 Reorder the terms: -3n + 6n2 = 14 Solving -3n + 6n2 = 14 Solving for variable 'n'. Reorder the terms: -14 + -3n + 6n2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + -3n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.333333333 + -0.5n + n2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -0.5n + 2.333333333 + n2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -0.5n + n2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -0.5n + n2 = 0 + 2.333333333 -0.5n + n2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -0.5n + n2 = 2.333333333 The n term is -0.5n. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5n + 0.0625 + n2 = 2.333333333 + 0.0625 Reorder the terms: 0.0625 + -0.5n + n2 = 2.333333333 + 0.0625 Combine like terms: 2.333333333 + 0.0625 = 2.395833333 0.0625 + -0.5n + n2 = 2.395833333 Factor a perfect square on the left side: (n + -0.25)(n + -0.25) = 2.395833333 Calculate the square root of the right side: 1.547847968 Break this problem into two subproblems by setting (n + -0.25) equal to 1.547847968 and -1.547847968.Subproblem 1
n + -0.25 = 1.547847968 Simplifying n + -0.25 = 1.547847968 Reorder the terms: -0.25 + n = 1.547847968 Solving -0.25 + n = 1.547847968 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = 1.547847968 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 1.547847968 + 0.25 n = 1.547847968 + 0.25 Combine like terms: 1.547847968 + 0.25 = 1.797847968 n = 1.797847968 Simplifying n = 1.797847968Subproblem 2
n + -0.25 = -1.547847968 Simplifying n + -0.25 = -1.547847968 Reorder the terms: -0.25 + n = -1.547847968 Solving -0.25 + n = -1.547847968 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = -1.547847968 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -1.547847968 + 0.25 n = -1.547847968 + 0.25 Combine like terms: -1.547847968 + 0.25 = -1.297847968 n = -1.297847968 Simplifying n = -1.297847968Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.797847968, -1.297847968}
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