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Simplifying 2n2 + 15n + 15 = 0 Reorder the terms: 15 + 15n + 2n2 = 0 Solving 15 + 15n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 7.5 + 7.5n + n2 = 0 Move the constant term to the right: Add '-7.5' to each side of the equation. 7.5 + 7.5n + -7.5 + n2 = 0 + -7.5 Reorder the terms: 7.5 + -7.5 + 7.5n + n2 = 0 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + 7.5n + n2 = 0 + -7.5 7.5n + n2 = 0 + -7.5 Combine like terms: 0 + -7.5 = -7.5 7.5n + n2 = -7.5 The n term is 7.5n. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5n + 14.0625 + n2 = -7.5 + 14.0625 Reorder the terms: 14.0625 + 7.5n + n2 = -7.5 + 14.0625 Combine like terms: -7.5 + 14.0625 = 6.5625 14.0625 + 7.5n + n2 = 6.5625 Factor a perfect square on the left side: (n + 3.75)(n + 3.75) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (n + 3.75) equal to 2.561737691 and -2.561737691.Subproblem 1
n + 3.75 = 2.561737691 Simplifying n + 3.75 = 2.561737691 Reorder the terms: 3.75 + n = 2.561737691 Solving 3.75 + n = 2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + n = 2.561737691 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + n = 2.561737691 + -3.75 n = 2.561737691 + -3.75 Combine like terms: 2.561737691 + -3.75 = -1.188262309 n = -1.188262309 Simplifying n = -1.188262309Subproblem 2
n + 3.75 = -2.561737691 Simplifying n + 3.75 = -2.561737691 Reorder the terms: 3.75 + n = -2.561737691 Solving 3.75 + n = -2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + n = -2.561737691 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + n = -2.561737691 + -3.75 n = -2.561737691 + -3.75 Combine like terms: -2.561737691 + -3.75 = -6.311737691 n = -6.311737691 Simplifying n = -6.311737691Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.188262309, -6.311737691}
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