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Simplifying 5n2 + 7n = -1 Reorder the terms: 7n + 5n2 = -1 Solving 7n + 5n2 = -1 Solving for variable 'n'. Reorder the terms: 1 + 7n + 5n2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + 7n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + 1.4n + n2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + 1.4n + -0.2 + n2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + 1.4n + n2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + 1.4n + n2 = 0 + -0.2 1.4n + n2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 1.4n + n2 = -0.2 The n term is 1.4n. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4n + 0.49 + n2 = -0.2 + 0.49 Reorder the terms: 0.49 + 1.4n + n2 = -0.2 + 0.49 Combine like terms: -0.2 + 0.49 = 0.29 0.49 + 1.4n + n2 = 0.29 Factor a perfect square on the left side: (n + 0.7)(n + 0.7) = 0.29 Calculate the square root of the right side: 0.538516481 Break this problem into two subproblems by setting (n + 0.7) equal to 0.538516481 and -0.538516481.Subproblem 1
n + 0.7 = 0.538516481 Simplifying n + 0.7 = 0.538516481 Reorder the terms: 0.7 + n = 0.538516481 Solving 0.7 + n = 0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = 0.538516481 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = 0.538516481 + -0.7 n = 0.538516481 + -0.7 Combine like terms: 0.538516481 + -0.7 = -0.161483519 n = -0.161483519 Simplifying n = -0.161483519Subproblem 2
n + 0.7 = -0.538516481 Simplifying n + 0.7 = -0.538516481 Reorder the terms: 0.7 + n = -0.538516481 Solving 0.7 + n = -0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = -0.538516481 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = -0.538516481 + -0.7 n = -0.538516481 + -0.7 Combine like terms: -0.538516481 + -0.7 = -1.238516481 n = -1.238516481 Simplifying n = -1.238516481Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.161483519, -1.238516481}
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