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Simplifying 10n2 + -9 = 5n Reorder the terms: -9 + 10n2 = 5n Solving -9 + 10n2 = 5n Solving for variable 'n'. Reorder the terms: -9 + -5n + 10n2 = 5n + -5n Combine like terms: 5n + -5n = 0 -9 + -5n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.9 + -0.5n + n2 = 0 Move the constant term to the right: Add '0.9' to each side of the equation. -0.9 + -0.5n + 0.9 + n2 = 0 + 0.9 Reorder the terms: -0.9 + 0.9 + -0.5n + n2 = 0 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + -0.5n + n2 = 0 + 0.9 -0.5n + n2 = 0 + 0.9 Combine like terms: 0 + 0.9 = 0.9 -0.5n + n2 = 0.9 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 = 0.9 + 0.0625 Reorder the terms: 0.0625 + -0.5n + n2 = 0.9 + 0.0625 Combine like terms: 0.9 + 0.0625 = 0.9625 0.0625 + -0.5n + n2 = 0.9625 Factor a perfect square on the left side: (n + -0.25)(n + -0.25) = 0.9625 Calculate the square root of the right side: 0.981070844 Break this problem into two subproblems by setting (n + -0.25) equal to 0.981070844 and -0.981070844.Subproblem 1
n + -0.25 = 0.981070844 Simplifying n + -0.25 = 0.981070844 Reorder the terms: -0.25 + n = 0.981070844 Solving -0.25 + n = 0.981070844 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 = 0.981070844 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 0.981070844 + 0.25 n = 0.981070844 + 0.25 Combine like terms: 0.981070844 + 0.25 = 1.231070844 n = 1.231070844 Simplifying n = 1.231070844Subproblem 2
n + -0.25 = -0.981070844 Simplifying n + -0.25 = -0.981070844 Reorder the terms: -0.25 + n = -0.981070844 Solving -0.25 + n = -0.981070844 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 = -0.981070844 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -0.981070844 + 0.25 n = -0.981070844 + 0.25 Combine like terms: -0.981070844 + 0.25 = -0.731070844 n = -0.731070844 Simplifying n = -0.731070844Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.231070844, -0.731070844}
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