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Simplifying -4n(5 + 5n) + -1 = -21 + -7n (5 * -4n + 5n * -4n) + -1 = -21 + -7n (-20n + -20n2) + -1 = -21 + -7n Reorder the terms: -1 + -20n + -20n2 = -21 + -7n Solving -1 + -20n + -20n2 = -21 + -7n Solving for variable 'n'. Reorder the terms: -1 + 21 + -20n + 7n + -20n2 = -21 + -7n + 21 + 7n Combine like terms: -1 + 21 = 20 20 + -20n + 7n + -20n2 = -21 + -7n + 21 + 7n Combine like terms: -20n + 7n = -13n 20 + -13n + -20n2 = -21 + -7n + 21 + 7n Reorder the terms: 20 + -13n + -20n2 = -21 + 21 + -7n + 7n Combine like terms: -21 + 21 = 0 20 + -13n + -20n2 = 0 + -7n + 7n 20 + -13n + -20n2 = -7n + 7n Combine like terms: -7n + 7n = 0 20 + -13n + -20n2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -1 + 0.65n + n2 = 0.0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.65n + 1 + n2 = 0.0 + 1 Reorder the terms: -1 + 1 + 0.65n + n2 = 0.0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.65n + n2 = 0.0 + 1 0.65n + n2 = 0.0 + 1 Combine like terms: 0.0 + 1 = 1 0.65n + n2 = 1 The n term is 0.65n. Take half its coefficient (0.325). Square it (0.105625) and add it to both sides. Add '0.105625' to each side of the equation. 0.65n + 0.105625 + n2 = 1 + 0.105625 Reorder the terms: 0.105625 + 0.65n + n2 = 1 + 0.105625 Combine like terms: 1 + 0.105625 = 1.105625 0.105625 + 0.65n + n2 = 1.105625 Factor a perfect square on the left side: (n + 0.325)(n + 0.325) = 1.105625 Calculate the square root of the right side: 1.051487042 Break this problem into two subproblems by setting (n + 0.325) equal to 1.051487042 and -1.051487042.Subproblem 1
n + 0.325 = 1.051487042 Simplifying n + 0.325 = 1.051487042 Reorder the terms: 0.325 + n = 1.051487042 Solving 0.325 + n = 1.051487042 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.325' to each side of the equation. 0.325 + -0.325 + n = 1.051487042 + -0.325 Combine like terms: 0.325 + -0.325 = 0.000 0.000 + n = 1.051487042 + -0.325 n = 1.051487042 + -0.325 Combine like terms: 1.051487042 + -0.325 = 0.726487042 n = 0.726487042 Simplifying n = 0.726487042Subproblem 2
n + 0.325 = -1.051487042 Simplifying n + 0.325 = -1.051487042 Reorder the terms: 0.325 + n = -1.051487042 Solving 0.325 + n = -1.051487042 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.325' to each side of the equation. 0.325 + -0.325 + n = -1.051487042 + -0.325 Combine like terms: 0.325 + -0.325 = 0.000 0.000 + n = -1.051487042 + -0.325 n = -1.051487042 + -0.325 Combine like terms: -1.051487042 + -0.325 = -1.376487042 n = -1.376487042 Simplifying n = -1.376487042Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.726487042, -1.376487042}
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