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Simplifying -10n + -7 = 2n2 Reorder the terms: -7 + -10n = 2n2 Solving -7 + -10n = 2n2 Solving for variable 'n'. Combine like terms: 2n2 + -2n2 = 0 -7 + -10n + -2n2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(7 + 10n + 2n2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(7 + 10n + 2n2)' equal to zero and attempt to solve: Simplifying 7 + 10n + 2n2 = 0 Solving 7 + 10n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + 5n + n2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 5n + -3.5 + n2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 5n + n2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 5n + n2 = 0 + -3.5 5n + n2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 5n + n2 = -3.5 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = -3.5 + 6.25 Reorder the terms: 6.25 + 5n + n2 = -3.5 + 6.25 Combine like terms: -3.5 + 6.25 = 2.75 6.25 + 5n + n2 = 2.75 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (n + 2.5) equal to 1.658312395 and -1.658312395.Subproblem 1
n + 2.5 = 1.658312395 Simplifying n + 2.5 = 1.658312395 Reorder the terms: 2.5 + n = 1.658312395 Solving 2.5 + n = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 1.658312395 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 1.658312395 + -2.5 n = 1.658312395 + -2.5 Combine like terms: 1.658312395 + -2.5 = -0.841687605 n = -0.841687605 Simplifying n = -0.841687605Subproblem 2
n + 2.5 = -1.658312395 Simplifying n + 2.5 = -1.658312395 Reorder the terms: 2.5 + n = -1.658312395 Solving 2.5 + n = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -1.658312395 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -1.658312395 + -2.5 n = -1.658312395 + -2.5 Combine like terms: -1.658312395 + -2.5 = -4.158312395 n = -4.158312395 Simplifying n = -4.158312395Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.841687605, -4.158312395}Solution
n = {-0.841687605, -4.158312395}
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