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Simplifying 2n2 + 5n + -9 = 0 Reorder the terms: -9 + 5n + 2n2 = 0 Solving -9 + 5n + 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'. -4.5 + 2.5n + n2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 2.5n + 4.5 + n2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 2.5n + n2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 2.5n + n2 = 0 + 4.5 2.5n + n2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 2.5n + n2 = 4.5 The n term is 2.5n. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5n + 1.5625 + n2 = 4.5 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = 4.5 + 1.5625 Combine like terms: 4.5 + 1.5625 = 6.0625 1.5625 + 2.5n + n2 = 6.0625 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (n + 1.25) equal to 2.46221445 and -2.46221445.Subproblem 1
n + 1.25 = 2.46221445 Simplifying n + 1.25 = 2.46221445 Reorder the terms: 1.25 + n = 2.46221445 Solving 1.25 + n = 2.46221445 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 2.46221445 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 2.46221445 + -1.25 n = 2.46221445 + -1.25 Combine like terms: 2.46221445 + -1.25 = 1.21221445 n = 1.21221445 Simplifying n = 1.21221445Subproblem 2
n + 1.25 = -2.46221445 Simplifying n + 1.25 = -2.46221445 Reorder the terms: 1.25 + n = -2.46221445 Solving 1.25 + n = -2.46221445 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -2.46221445 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -2.46221445 + -1.25 n = -2.46221445 + -1.25 Combine like terms: -2.46221445 + -1.25 = -3.71221445 n = -3.71221445 Simplifying n = -3.71221445Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.21221445, -3.71221445}
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