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Simplifying (4n + 1)(2n + -3) = n Reorder the terms: (1 + 4n)(2n + -3) = n Reorder the terms: (1 + 4n)(-3 + 2n) = n Multiply (1 + 4n) * (-3 + 2n) (1(-3 + 2n) + 4n * (-3 + 2n)) = n ((-3 * 1 + 2n * 1) + 4n * (-3 + 2n)) = n ((-3 + 2n) + 4n * (-3 + 2n)) = n (-3 + 2n + (-3 * 4n + 2n * 4n)) = n (-3 + 2n + (-12n + 8n2)) = n Combine like terms: 2n + -12n = -10n (-3 + -10n + 8n2) = n Solving -3 + -10n + 8n2 = n Solving for variable 'n'. Reorder the terms: -3 + -10n + -1n + 8n2 = n + -1n Combine like terms: -10n + -1n = -11n -3 + -11n + 8n2 = n + -1n Combine like terms: n + -1n = 0 -3 + -11n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + -1.375n + n2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -1.375n + 0.375 + n2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -1.375n + n2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -1.375n + n2 = 0 + 0.375 -1.375n + n2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -1.375n + n2 = 0.375 The n term is -1.375n. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375n + 0.47265625 + n2 = 0.375 + 0.47265625 Reorder the terms: 0.47265625 + -1.375n + n2 = 0.375 + 0.47265625 Combine like terms: 0.375 + 0.47265625 = 0.84765625 0.47265625 + -1.375n + n2 = 0.84765625 Factor a perfect square on the left side: (n + -0.6875)(n + -0.6875) = 0.84765625 Calculate the square root of the right side: 0.920682491 Break this problem into two subproblems by setting (n + -0.6875) equal to 0.920682491 and -0.920682491.Subproblem 1
n + -0.6875 = 0.920682491 Simplifying n + -0.6875 = 0.920682491 Reorder the terms: -0.6875 + n = 0.920682491 Solving -0.6875 + n = 0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + n = 0.920682491 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + n = 0.920682491 + 0.6875 n = 0.920682491 + 0.6875 Combine like terms: 0.920682491 + 0.6875 = 1.608182491 n = 1.608182491 Simplifying n = 1.608182491Subproblem 2
n + -0.6875 = -0.920682491 Simplifying n + -0.6875 = -0.920682491 Reorder the terms: -0.6875 + n = -0.920682491 Solving -0.6875 + n = -0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + n = -0.920682491 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + n = -0.920682491 + 0.6875 n = -0.920682491 + 0.6875 Combine like terms: -0.920682491 + 0.6875 = -0.233182491 n = -0.233182491 Simplifying n = -0.233182491Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.608182491, -0.233182491}
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