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Simplifying -9n2 + -1n + 6 = 7 Reorder the terms: 6 + -1n + -9n2 = 7 Solving 6 + -1n + -9n2 = 7 Solving for variable 'n'. Reorder the terms: 6 + -7 + -1n + -9n2 = 7 + -7 Combine like terms: 6 + -7 = -1 -1 + -1n + -9n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -1 + -1n + -9n2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + n + 9n2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + n + 9n2)' equal to zero and attempt to solve: Simplifying 1 + n + 9n2 = 0 Solving 1 + n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.1111111111 + 0.1111111111n + n2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + 0.1111111111n + -0.1111111111 + n2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + 0.1111111111n + n2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + 0.1111111111n + n2 = 0 + -0.1111111111 0.1111111111n + n2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 0.1111111111n + n2 = -0.1111111111 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1111111111n + 0.25 + n2 = -0.1111111111 + 0.25 Reorder the terms: 0.25 + 0.1111111111n + n2 = -0.1111111111 + 0.25 Combine like terms: -0.1111111111 + 0.25 = 0.1388888889 0.25 + 0.1111111111n + n2 = 0.1388888889 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.1388888889 Calculate the square root of the right side: 0.372677996 Break this problem into two subproblems by setting (n + 0.5) equal to 0.372677996 and -0.372677996.Subproblem 1
n + 0.5 = 0.372677996 Simplifying n + 0.5 = 0.372677996 Reorder the terms: 0.5 + n = 0.372677996 Solving 0.5 + n = 0.372677996 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.372677996 + -0.5 n = 0.372677996 + -0.5 Combine like terms: 0.372677996 + -0.5 = -0.127322004 n = -0.127322004 Simplifying n = -0.127322004Subproblem 2
n + 0.5 = -0.372677996 Simplifying n + 0.5 = -0.372677996 Reorder the terms: 0.5 + n = -0.372677996 Solving 0.5 + n = -0.372677996 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.372677996 + -0.5 n = -0.372677996 + -0.5 Combine like terms: -0.372677996 + -0.5 = -0.872677996 n = -0.872677996 Simplifying n = -0.872677996Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.127322004, -0.872677996}Solution
n = {-0.127322004, -0.872677996}
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