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Simplifying 72n4 + -129n2 + 50 = 0 Reorder the terms: 50 + -129n2 + 72n4 = 0 Solving 50 + -129n2 + 72n4 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 72 the coefficient of the squared term: Divide each side by '72'. 0.6944444444 + -1.791666667n2 + n4 = 0 Move the constant term to the right: Add '-0.6944444444' to each side of the equation. 0.6944444444 + -1.791666667n2 + -0.6944444444 + n4 = 0 + -0.6944444444 Reorder the terms: 0.6944444444 + -0.6944444444 + -1.791666667n2 + n4 = 0 + -0.6944444444 Combine like terms: 0.6944444444 + -0.6944444444 = 0.0000000000 0.0000000000 + -1.791666667n2 + n4 = 0 + -0.6944444444 -1.791666667n2 + n4 = 0 + -0.6944444444 Combine like terms: 0 + -0.6944444444 = -0.6944444444 -1.791666667n2 + n4 = -0.6944444444 The n term is -1.791666667n2. Take half its coefficient (-0.8958333335). Square it (0.8025173614) and add it to both sides. Add '0.8025173614' to each side of the equation. -1.791666667n2 + 0.8025173614 + n4 = -0.6944444444 + 0.8025173614 Reorder the terms: 0.8025173614 + -1.791666667n2 + n4 = -0.6944444444 + 0.8025173614 Combine like terms: -0.6944444444 + 0.8025173614 = 0.108072917 0.8025173614 + -1.791666667n2 + n4 = 0.108072917 Factor a perfect square on the left side: (n2 + -0.8958333335)(n2 + -0.8958333335) = 0.108072917 Calculate the square root of the right side: 0.328744455 Break this problem into two subproblems by setting (n2 + -0.8958333335) equal to 0.328744455 and -0.328744455.Subproblem 1
n2 + -0.8958333335 = 0.328744455 Simplifying n2 + -0.8958333335 = 0.328744455 Reorder the terms: -0.8958333335 + n2 = 0.328744455 Solving -0.8958333335 + n2 = 0.328744455 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8958333335' to each side of the equation. -0.8958333335 + 0.8958333335 + n2 = 0.328744455 + 0.8958333335 Combine like terms: -0.8958333335 + 0.8958333335 = 0.0000000000 0.0000000000 + n2 = 0.328744455 + 0.8958333335 n2 = 0.328744455 + 0.8958333335 Combine like terms: 0.328744455 + 0.8958333335 = 1.2245777885 n2 = 1.2245777885 Simplifying n2 = 1.2245777885 Take the square root of each side: n = {-1.106606429, 1.106606429}Subproblem 2
n2 + -0.8958333335 = -0.328744455 Simplifying n2 + -0.8958333335 = -0.328744455 Reorder the terms: -0.8958333335 + n2 = -0.328744455 Solving -0.8958333335 + n2 = -0.328744455 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8958333335' to each side of the equation. -0.8958333335 + 0.8958333335 + n2 = -0.328744455 + 0.8958333335 Combine like terms: -0.8958333335 + 0.8958333335 = 0.0000000000 0.0000000000 + n2 = -0.328744455 + 0.8958333335 n2 = -0.328744455 + 0.8958333335 Combine like terms: -0.328744455 + 0.8958333335 = 0.5670888785 n2 = 0.5670888785 Simplifying n2 = 0.5670888785 Take the square root of each side: n = {-0.753053038, 0.753053038}Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.106606429, 1.106606429, -0.753053038, 0.753053038}
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