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Simplifying 3n2 + 5 = -9n Reorder the terms: 5 + 3n2 = -9n Solving 5 + 3n2 = -9n Solving for variable 'n'. Reorder the terms: 5 + 9n + 3n2 = -9n + 9n Combine like terms: -9n + 9n = 0 5 + 9n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.666666667 + 3n + n2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 3n + -1.666666667 + n2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 3n + n2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 3n + n2 = 0 + -1.666666667 3n + n2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 3n + n2 = -1.666666667 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = -1.666666667 + 2.25 Reorder the terms: 2.25 + 3n + n2 = -1.666666667 + 2.25 Combine like terms: -1.666666667 + 2.25 = 0.583333333 2.25 + 3n + n2 = 0.583333333 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 0.583333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (n + 1.5) equal to 0.763762616 and -0.763762616.Subproblem 1
n + 1.5 = 0.763762616 Simplifying n + 1.5 = 0.763762616 Reorder the terms: 1.5 + n = 0.763762616 Solving 1.5 + n = 0.763762616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 0.763762616 + -1.5 n = 0.763762616 + -1.5 Combine like terms: 0.763762616 + -1.5 = -0.736237384 n = -0.736237384 Simplifying n = -0.736237384Subproblem 2
n + 1.5 = -0.763762616 Simplifying n + 1.5 = -0.763762616 Reorder the terms: 1.5 + n = -0.763762616 Solving 1.5 + n = -0.763762616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -0.763762616 + -1.5 n = -0.763762616 + -1.5 Combine like terms: -0.763762616 + -1.5 = -2.263762616 n = -2.263762616 Simplifying n = -2.263762616Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.736237384, -2.263762616}
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