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Simplifying 4n + -1(n + -65) * n = 100 Reorder the terms: 4n + -1(-65 + n) * n = 100 Reorder the terms for easier multiplication: 4n + -1n(-65 + n) = 100 4n + (-65 * -1n + n * -1n) = 100 4n + (65n + -1n2) = 100 Combine like terms: 4n + 65n = 69n 69n + -1n2 = 100 Solving 69n + -1n2 = 100 Solving for variable 'n'. Reorder the terms: -100 + 69n + -1n2 = 100 + -100 Combine like terms: 100 + -100 = 0 -100 + 69n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 100 + -69n + n2 = 0 Move the constant term to the right: Add '-100' to each side of the equation. 100 + -69n + -100 + n2 = 0 + -100 Reorder the terms: 100 + -100 + -69n + n2 = 0 + -100 Combine like terms: 100 + -100 = 0 0 + -69n + n2 = 0 + -100 -69n + n2 = 0 + -100 Combine like terms: 0 + -100 = -100 -69n + n2 = -100 The n term is -69n. Take half its coefficient (-34.5). Square it (1190.25) and add it to both sides. Add '1190.25' to each side of the equation. -69n + 1190.25 + n2 = -100 + 1190.25 Reorder the terms: 1190.25 + -69n + n2 = -100 + 1190.25 Combine like terms: -100 + 1190.25 = 1090.25 1190.25 + -69n + n2 = 1090.25 Factor a perfect square on the left side: (n + -34.5)(n + -34.5) = 1090.25 Calculate the square root of the right side: 33.018933962 Break this problem into two subproblems by setting (n + -34.5) equal to 33.018933962 and -33.018933962.Subproblem 1
n + -34.5 = 33.018933962 Simplifying n + -34.5 = 33.018933962 Reorder the terms: -34.5 + n = 33.018933962 Solving -34.5 + n = 33.018933962 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '34.5' to each side of the equation. -34.5 + 34.5 + n = 33.018933962 + 34.5 Combine like terms: -34.5 + 34.5 = 0.0 0.0 + n = 33.018933962 + 34.5 n = 33.018933962 + 34.5 Combine like terms: 33.018933962 + 34.5 = 67.518933962 n = 67.518933962 Simplifying n = 67.518933962Subproblem 2
n + -34.5 = -33.018933962 Simplifying n + -34.5 = -33.018933962 Reorder the terms: -34.5 + n = -33.018933962 Solving -34.5 + n = -33.018933962 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '34.5' to each side of the equation. -34.5 + 34.5 + n = -33.018933962 + 34.5 Combine like terms: -34.5 + 34.5 = 0.0 0.0 + n = -33.018933962 + 34.5 n = -33.018933962 + 34.5 Combine like terms: -33.018933962 + 34.5 = 1.481066038 n = 1.481066038 Simplifying n = 1.481066038Solution
The solution to the problem is based on the solutions from the subproblems. n = {67.518933962, 1.481066038}
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