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Simplifying 2n2 = -10n + -5 Reorder the terms: 2n2 = -5 + -10n Solving 2n2 = -5 + -10n Solving for variable 'n'. Reorder the terms: 5 + 10n + 2n2 = -5 + -10n + 5 + 10n Reorder the terms: 5 + 10n + 2n2 = -5 + 5 + -10n + 10n Combine like terms: -5 + 5 = 0 5 + 10n + 2n2 = 0 + -10n + 10n 5 + 10n + 2n2 = -10n + 10n Combine like terms: -10n + 10n = 0 5 + 10n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + 5n + n2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 5n + -2.5 + n2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 5n + n2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 5n + n2 = 0 + -2.5 5n + n2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 5n + n2 = -2.5 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = -2.5 + 6.25 Reorder the terms: 6.25 + 5n + n2 = -2.5 + 6.25 Combine like terms: -2.5 + 6.25 = 3.75 6.25 + 5n + n2 = 3.75 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (n + 2.5) equal to 1.936491673 and -1.936491673.Subproblem 1
n + 2.5 = 1.936491673 Simplifying n + 2.5 = 1.936491673 Reorder the terms: 2.5 + n = 1.936491673 Solving 2.5 + n = 1.936491673 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 1.936491673 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 1.936491673 + -2.5 n = 1.936491673 + -2.5 Combine like terms: 1.936491673 + -2.5 = -0.563508327 n = -0.563508327 Simplifying n = -0.563508327Subproblem 2
n + 2.5 = -1.936491673 Simplifying n + 2.5 = -1.936491673 Reorder the terms: 2.5 + n = -1.936491673 Solving 2.5 + n = -1.936491673 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -1.936491673 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -1.936491673 + -2.5 n = -1.936491673 + -2.5 Combine like terms: -1.936491673 + -2.5 = -4.436491673 n = -4.436491673 Simplifying n = -4.436491673Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.563508327, -4.436491673}
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