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Simplifying n2 + 6n + 7n = -5 Reorder the terms: 6n + 7n + n2 = -5 Combine like terms: 6n + 7n = 13n 13n + n2 = -5 Solving 13n + n2 = -5 Solving for variable 'n'. Reorder the terms: 5 + 13n + n2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 13n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 13n + -5 + n2 = 0 + -5 Reorder the terms: 5 + -5 + 13n + n2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 13n + n2 = 0 + -5 13n + n2 = 0 + -5 Combine like terms: 0 + -5 = -5 13n + n2 = -5 The n term is 13n. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13n + 42.25 + n2 = -5 + 42.25 Reorder the terms: 42.25 + 13n + n2 = -5 + 42.25 Combine like terms: -5 + 42.25 = 37.25 42.25 + 13n + n2 = 37.25 Factor a perfect square on the left side: (n + 6.5)(n + 6.5) = 37.25 Calculate the square root of the right side: 6.103277808 Break this problem into two subproblems by setting (n + 6.5) equal to 6.103277808 and -6.103277808.Subproblem 1
n + 6.5 = 6.103277808 Simplifying n + 6.5 = 6.103277808 Reorder the terms: 6.5 + n = 6.103277808 Solving 6.5 + n = 6.103277808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = 6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 6.103277808 + -6.5 n = 6.103277808 + -6.5 Combine like terms: 6.103277808 + -6.5 = -0.396722192 n = -0.396722192 Simplifying n = -0.396722192Subproblem 2
n + 6.5 = -6.103277808 Simplifying n + 6.5 = -6.103277808 Reorder the terms: 6.5 + n = -6.103277808 Solving 6.5 + n = -6.103277808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = -6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -6.103277808 + -6.5 n = -6.103277808 + -6.5 Combine like terms: -6.103277808 + -6.5 = -12.603277808 n = -12.603277808 Simplifying n = -12.603277808Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.396722192, -12.603277808}
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