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Simplifying -1n2 + -3n + 20 = 0 Reorder the terms: 20 + -3n + -1n2 = 0 Solving 20 + -3n + -1n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -20 + 3n + n2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + 3n + 20 + n2 = 0 + 20 Reorder the terms: -20 + 20 + 3n + n2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 3n + n2 = 0 + 20 3n + n2 = 0 + 20 Combine like terms: 0 + 20 = 20 3n + n2 = 20 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 20 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 20 + 2.25 Combine like terms: 20 + 2.25 = 22.25 2.25 + 3n + n2 = 22.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (n + 1.5) equal to 4.716990566 and -4.716990566.Subproblem 1
n + 1.5 = 4.716990566 Simplifying n + 1.5 = 4.716990566 Reorder the terms: 1.5 + n = 4.716990566 Solving 1.5 + n = 4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.716990566 + -1.5 n = 4.716990566 + -1.5 Combine like terms: 4.716990566 + -1.5 = 3.216990566 n = 3.216990566 Simplifying n = 3.216990566Subproblem 2
n + 1.5 = -4.716990566 Simplifying n + 1.5 = -4.716990566 Reorder the terms: 1.5 + n = -4.716990566 Solving 1.5 + n = -4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.716990566 + -1.5 n = -4.716990566 + -1.5 Combine like terms: -4.716990566 + -1.5 = -6.216990566 n = -6.216990566 Simplifying n = -6.216990566Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.216990566, -6.216990566}
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