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Simplifying -4n2 + n + 10 = 0 Reorder the terms: 10 + n + -4n2 = 0 Solving 10 + n + -4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.5 + -0.25n + n2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -0.25n + 2.5 + n2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -0.25n + n2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -0.25n + n2 = 0 + 2.5 -0.25n + n2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -0.25n + n2 = 2.5 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. -0.25n + 0.25 + n2 = 2.5 + 0.25 Reorder the terms: 0.25 + -0.25n + n2 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 0.25 + -0.25n + n2 = 2.75 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (n + 0.5) equal to 1.658312395 and -1.658312395.Subproblem 1
n + 0.5 = 1.658312395 Simplifying n + 0.5 = 1.658312395 Reorder the terms: 0.5 + n = 1.658312395 Solving 0.5 + n = 1.658312395 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 = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.658312395 + -0.5 n = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 n = 1.158312395 Simplifying n = 1.158312395Subproblem 2
n + 0.5 = -1.658312395 Simplifying n + 0.5 = -1.658312395 Reorder the terms: 0.5 + n = -1.658312395 Solving 0.5 + n = -1.658312395 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 = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.658312395 + -0.5 n = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 n = -2.158312395 Simplifying n = -2.158312395Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.158312395, -2.158312395}
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