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Simplifying 5n2 + -5 = -4n Reorder the terms: -5 + 5n2 = -4n Solving -5 + 5n2 = -4n Solving for variable 'n'. Reorder the terms: -5 + 4n + 5n2 = -4n + 4n Combine like terms: -4n + 4n = 0 -5 + 4n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1 + 0.8n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.8n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + 0.8n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.8n + n2 = 0 + 1 0.8n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.8n + n2 = 1 The n term is 0.8n. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8n + 0.16 + n2 = 1 + 0.16 Reorder the terms: 0.16 + 0.8n + n2 = 1 + 0.16 Combine like terms: 1 + 0.16 = 1.16 0.16 + 0.8n + n2 = 1.16 Factor a perfect square on the left side: (n + 0.4)(n + 0.4) = 1.16 Calculate the square root of the right side: 1.077032961 Break this problem into two subproblems by setting (n + 0.4) equal to 1.077032961 and -1.077032961.Subproblem 1
n + 0.4 = 1.077032961 Simplifying n + 0.4 = 1.077032961 Reorder the terms: 0.4 + n = 1.077032961 Solving 0.4 + n = 1.077032961 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + n = 1.077032961 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = 1.077032961 + -0.4 n = 1.077032961 + -0.4 Combine like terms: 1.077032961 + -0.4 = 0.677032961 n = 0.677032961 Simplifying n = 0.677032961Subproblem 2
n + 0.4 = -1.077032961 Simplifying n + 0.4 = -1.077032961 Reorder the terms: 0.4 + n = -1.077032961 Solving 0.4 + n = -1.077032961 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + n = -1.077032961 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = -1.077032961 + -0.4 n = -1.077032961 + -0.4 Combine like terms: -1.077032961 + -0.4 = -1.477032961 n = -1.477032961 Simplifying n = -1.477032961Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.677032961, -1.477032961}
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