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Simplifying n2 + 14n + -78 = -4 Reorder the terms: -78 + 14n + n2 = -4 Solving -78 + 14n + n2 = -4 Solving for variable 'n'. Reorder the terms: -78 + 4 + 14n + n2 = -4 + 4 Combine like terms: -78 + 4 = -74 -74 + 14n + n2 = -4 + 4 Combine like terms: -4 + 4 = 0 -74 + 14n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '74' to each side of the equation. -74 + 14n + 74 + n2 = 0 + 74 Reorder the terms: -74 + 74 + 14n + n2 = 0 + 74 Combine like terms: -74 + 74 = 0 0 + 14n + n2 = 0 + 74 14n + n2 = 0 + 74 Combine like terms: 0 + 74 = 74 14n + n2 = 74 The n term is 14n. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14n + 49 + n2 = 74 + 49 Reorder the terms: 49 + 14n + n2 = 74 + 49 Combine like terms: 74 + 49 = 123 49 + 14n + n2 = 123 Factor a perfect square on the left side: (n + 7)(n + 7) = 123 Calculate the square root of the right side: 11.090536506 Break this problem into two subproblems by setting (n + 7) equal to 11.090536506 and -11.090536506.Subproblem 1
n + 7 = 11.090536506 Simplifying n + 7 = 11.090536506 Reorder the terms: 7 + n = 11.090536506 Solving 7 + n = 11.090536506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = 11.090536506 + -7 Combine like terms: 7 + -7 = 0 0 + n = 11.090536506 + -7 n = 11.090536506 + -7 Combine like terms: 11.090536506 + -7 = 4.090536506 n = 4.090536506 Simplifying n = 4.090536506Subproblem 2
n + 7 = -11.090536506 Simplifying n + 7 = -11.090536506 Reorder the terms: 7 + n = -11.090536506 Solving 7 + n = -11.090536506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = -11.090536506 + -7 Combine like terms: 7 + -7 = 0 0 + n = -11.090536506 + -7 n = -11.090536506 + -7 Combine like terms: -11.090536506 + -7 = -18.090536506 n = -18.090536506 Simplifying n = -18.090536506Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.090536506, -18.090536506}
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