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Simplifying n2 + 3n + -10 = -5n2 + -3 Reorder the terms: -10 + 3n + n2 = -5n2 + -3 Reorder the terms: -10 + 3n + n2 = -3 + -5n2 Solving -10 + 3n + n2 = -3 + -5n2 Solving for variable 'n'. Reorder the terms: -10 + 3 + 3n + n2 + 5n2 = -3 + -5n2 + 3 + 5n2 Combine like terms: -10 + 3 = -7 -7 + 3n + n2 + 5n2 = -3 + -5n2 + 3 + 5n2 Combine like terms: n2 + 5n2 = 6n2 -7 + 3n + 6n2 = -3 + -5n2 + 3 + 5n2 Reorder the terms: -7 + 3n + 6n2 = -3 + 3 + -5n2 + 5n2 Combine like terms: -3 + 3 = 0 -7 + 3n + 6n2 = 0 + -5n2 + 5n2 -7 + 3n + 6n2 = -5n2 + 5n2 Combine like terms: -5n2 + 5n2 = 0 -7 + 3n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + 0.5n + n2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 0.5n + 1.166666667 + n2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 0.5n + n2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 0.5n + n2 = 0 + 1.166666667 0.5n + n2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 0.5n + n2 = 1.166666667 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 1.166666667 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 1.166666667 + 0.0625 Combine like terms: 1.166666667 + 0.0625 = 1.229166667 0.0625 + 0.5n + n2 = 1.229166667 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 1.229166667 Calculate the square root of the right side: 1.108677891 Break this problem into two subproblems by setting (n + 0.25) equal to 1.108677891 and -1.108677891.Subproblem 1
n + 0.25 = 1.108677891 Simplifying n + 0.25 = 1.108677891 Reorder the terms: 0.25 + n = 1.108677891 Solving 0.25 + n = 1.108677891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.108677891 + -0.25 n = 1.108677891 + -0.25 Combine like terms: 1.108677891 + -0.25 = 0.858677891 n = 0.858677891 Simplifying n = 0.858677891Subproblem 2
n + 0.25 = -1.108677891 Simplifying n + 0.25 = -1.108677891 Reorder the terms: 0.25 + n = -1.108677891 Solving 0.25 + n = -1.108677891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.108677891 + -0.25 n = -1.108677891 + -0.25 Combine like terms: -1.108677891 + -0.25 = -1.358677891 n = -1.358677891 Simplifying n = -1.358677891Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.858677891, -1.358677891}
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