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Simplifying (5n2 + 2n + -1) = 0 Reorder the terms: (-1 + 2n + 5n2) = 0 Remove parenthesis around (-1 + 2n + 5n2) -1 + 2n + 5n2 = 0 Solving -1 + 2n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.2 + 0.4n + n2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.4n + 0.2 + n2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.4n + n2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.4n + n2 = 0 + 0.2 0.4n + n2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 0.4n + n2 = 0.2 The n term is 0.4n. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4n + 0.04 + n2 = 0.2 + 0.04 Reorder the terms: 0.04 + 0.4n + n2 = 0.2 + 0.04 Combine like terms: 0.2 + 0.04 = 0.24 0.04 + 0.4n + n2 = 0.24 Factor a perfect square on the left side: (n + 0.2)(n + 0.2) = 0.24 Calculate the square root of the right side: 0.489897949 Break this problem into two subproblems by setting (n + 0.2) equal to 0.489897949 and -0.489897949.Subproblem 1
n + 0.2 = 0.489897949 Simplifying n + 0.2 = 0.489897949 Reorder the terms: 0.2 + n = 0.489897949 Solving 0.2 + n = 0.489897949 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + n = 0.489897949 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + n = 0.489897949 + -0.2 n = 0.489897949 + -0.2 Combine like terms: 0.489897949 + -0.2 = 0.289897949 n = 0.289897949 Simplifying n = 0.289897949Subproblem 2
n + 0.2 = -0.489897949 Simplifying n + 0.2 = -0.489897949 Reorder the terms: 0.2 + n = -0.489897949 Solving 0.2 + n = -0.489897949 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + n = -0.489897949 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + n = -0.489897949 + -0.2 n = -0.489897949 + -0.2 Combine like terms: -0.489897949 + -0.2 = -0.689897949 n = -0.689897949 Simplifying n = -0.689897949Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.289897949, -0.689897949}
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