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Simplifying 10n2 + -5n + -10 = -6 Reorder the terms: -10 + -5n + 10n2 = -6 Solving -10 + -5n + 10n2 = -6 Solving for variable 'n'. Reorder the terms: -10 + 6 + -5n + 10n2 = -6 + 6 Combine like terms: -10 + 6 = -4 -4 + -5n + 10n2 = -6 + 6 Combine like terms: -6 + 6 = 0 -4 + -5n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.4 + -0.5n + n2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -0.5n + 0.4 + n2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -0.5n + n2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -0.5n + n2 = 0 + 0.4 -0.5n + n2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -0.5n + n2 = 0.4 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 = 0.4 + 0.0625 Reorder the terms: 0.0625 + -0.5n + n2 = 0.4 + 0.0625 Combine like terms: 0.4 + 0.0625 = 0.4625 0.0625 + -0.5n + n2 = 0.4625 Factor a perfect square on the left side: (n + -0.25)(n + -0.25) = 0.4625 Calculate the square root of the right side: 0.680073525 Break this problem into two subproblems by setting (n + -0.25) equal to 0.680073525 and -0.680073525.Subproblem 1
n + -0.25 = 0.680073525 Simplifying n + -0.25 = 0.680073525 Reorder the terms: -0.25 + n = 0.680073525 Solving -0.25 + n = 0.680073525 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 = 0.680073525 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 0.680073525 + 0.25 n = 0.680073525 + 0.25 Combine like terms: 0.680073525 + 0.25 = 0.930073525 n = 0.930073525 Simplifying n = 0.930073525Subproblem 2
n + -0.25 = -0.680073525 Simplifying n + -0.25 = -0.680073525 Reorder the terms: -0.25 + n = -0.680073525 Solving -0.25 + n = -0.680073525 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 = -0.680073525 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -0.680073525 + 0.25 n = -0.680073525 + 0.25 Combine like terms: -0.680073525 + 0.25 = -0.430073525 n = -0.430073525 Simplifying n = -0.430073525Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.930073525, -0.430073525}
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