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Simplifying 2n2 = 3n + 12 Reorder the terms: 2n2 = 12 + 3n Solving 2n2 = 12 + 3n Solving for variable 'n'. Reorder the terms: -12 + -3n + 2n2 = 12 + 3n + -12 + -3n Reorder the terms: -12 + -3n + 2n2 = 12 + -12 + 3n + -3n Combine like terms: 12 + -12 = 0 -12 + -3n + 2n2 = 0 + 3n + -3n -12 + -3n + 2n2 = 3n + -3n Combine like terms: 3n + -3n = 0 -12 + -3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6 + -1.5n + n2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -1.5n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + -1.5n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -1.5n + n2 = 0 + 6 -1.5n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 -1.5n + n2 = 6 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 6 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 6 + 0.5625 Combine like terms: 6 + 0.5625 = 6.5625 0.5625 + -1.5n + n2 = 6.5625 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (n + -0.75) equal to 2.561737691 and -2.561737691.Subproblem 1
n + -0.75 = 2.561737691 Simplifying n + -0.75 = 2.561737691 Reorder the terms: -0.75 + n = 2.561737691 Solving -0.75 + n = 2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 2.561737691 + 0.75 n = 2.561737691 + 0.75 Combine like terms: 2.561737691 + 0.75 = 3.311737691 n = 3.311737691 Simplifying n = 3.311737691Subproblem 2
n + -0.75 = -2.561737691 Simplifying n + -0.75 = -2.561737691 Reorder the terms: -0.75 + n = -2.561737691 Solving -0.75 + n = -2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -2.561737691 + 0.75 n = -2.561737691 + 0.75 Combine like terms: -2.561737691 + 0.75 = -1.811737691 n = -1.811737691 Simplifying n = -1.811737691Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.311737691, -1.811737691}
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