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Simplifying 2n2 = 7 + -7n Solving 2n2 = 7 + -7n Solving for variable 'n'. Reorder the terms: -7 + 7n + 2n2 = 7 + -7n + -7 + 7n Reorder the terms: -7 + 7n + 2n2 = 7 + -7 + -7n + 7n Combine like terms: 7 + -7 = 0 -7 + 7n + 2n2 = 0 + -7n + 7n -7 + 7n + 2n2 = -7n + 7n Combine like terms: -7n + 7n = 0 -7 + 7n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 3.5n + n2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 3.5n + 3.5 + n2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 3.5n + n2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 3.5n + n2 = 0 + 3.5 3.5n + n2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 3.5n + n2 = 3.5 The n term is 3.5n. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5n + 3.0625 + n2 = 3.5 + 3.0625 Reorder the terms: 3.0625 + 3.5n + n2 = 3.5 + 3.0625 Combine like terms: 3.5 + 3.0625 = 6.5625 3.0625 + 3.5n + n2 = 6.5625 Factor a perfect square on the left side: (n + 1.75)(n + 1.75) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (n + 1.75) equal to 2.561737691 and -2.561737691.Subproblem 1
n + 1.75 = 2.561737691 Simplifying n + 1.75 = 2.561737691 Reorder the terms: 1.75 + n = 2.561737691 Solving 1.75 + n = 2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 2.561737691 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 2.561737691 + -1.75 n = 2.561737691 + -1.75 Combine like terms: 2.561737691 + -1.75 = 0.811737691 n = 0.811737691 Simplifying n = 0.811737691Subproblem 2
n + 1.75 = -2.561737691 Simplifying n + 1.75 = -2.561737691 Reorder the terms: 1.75 + n = -2.561737691 Solving 1.75 + n = -2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -2.561737691 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -2.561737691 + -1.75 n = -2.561737691 + -1.75 Combine like terms: -2.561737691 + -1.75 = -4.311737691 n = -4.311737691 Simplifying n = -4.311737691Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.811737691, -4.311737691}
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