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Simplifying 16n2 + n + -25 = 0 Reorder the terms: -25 + n + 16n2 = 0 Solving -25 + n + 16n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.5625 + 0.0625n + n2 = 0 Move the constant term to the right: Add '1.5625' to each side of the equation. -1.5625 + 0.0625n + 1.5625 + n2 = 0 + 1.5625 Reorder the terms: -1.5625 + 1.5625 + 0.0625n + n2 = 0 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + 0.0625n + n2 = 0 + 1.5625 0.0625n + n2 = 0 + 1.5625 Combine like terms: 0 + 1.5625 = 1.5625 0.0625n + n2 = 1.5625 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.0625n + 0.25 + n2 = 1.5625 + 0.25 Reorder the terms: 0.25 + 0.0625n + n2 = 1.5625 + 0.25 Combine like terms: 1.5625 + 0.25 = 1.8125 0.25 + 0.0625n + n2 = 1.8125 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 1.8125 Calculate the square root of the right side: 1.346291202 Break this problem into two subproblems by setting (n + 0.5) equal to 1.346291202 and -1.346291202.Subproblem 1
n + 0.5 = 1.346291202 Simplifying n + 0.5 = 1.346291202 Reorder the terms: 0.5 + n = 1.346291202 Solving 0.5 + n = 1.346291202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.346291202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.346291202 + -0.5 n = 1.346291202 + -0.5 Combine like terms: 1.346291202 + -0.5 = 0.846291202 n = 0.846291202 Simplifying n = 0.846291202Subproblem 2
n + 0.5 = -1.346291202 Simplifying n + 0.5 = -1.346291202 Reorder the terms: 0.5 + n = -1.346291202 Solving 0.5 + n = -1.346291202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.346291202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.346291202 + -0.5 n = -1.346291202 + -0.5 Combine like terms: -1.346291202 + -0.5 = -1.846291202 n = -1.846291202 Simplifying n = -1.846291202Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.846291202, -1.846291202}
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