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Simplifying 3n2 + -9n = 2 Reorder the terms: -9n + 3n2 = 2 Solving -9n + 3n2 = 2 Solving for variable 'n'. Reorder the terms: -2 + -9n + 3n2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + -9n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.6666666667 + -3n + n2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -3n + 0.6666666667 + n2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -3n + n2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -3n + n2 = 0 + 0.6666666667 -3n + n2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -3n + n2 = 0.6666666667 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 = 0.6666666667 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 0.6666666667 + 2.25 Combine like terms: 0.6666666667 + 2.25 = 2.9166666667 2.25 + -3n + n2 = 2.9166666667 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 2.9166666667 Calculate the square root of the right side: 1.707825128 Break this problem into two subproblems by setting (n + -1.5) equal to 1.707825128 and -1.707825128.Subproblem 1
n + -1.5 = 1.707825128 Simplifying n + -1.5 = 1.707825128 Reorder the terms: -1.5 + n = 1.707825128 Solving -1.5 + n = 1.707825128 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 = 1.707825128 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.707825128 + 1.5 n = 1.707825128 + 1.5 Combine like terms: 1.707825128 + 1.5 = 3.207825128 n = 3.207825128 Simplifying n = 3.207825128Subproblem 2
n + -1.5 = -1.707825128 Simplifying n + -1.5 = -1.707825128 Reorder the terms: -1.5 + n = -1.707825128 Solving -1.5 + n = -1.707825128 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 = -1.707825128 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.707825128 + 1.5 n = -1.707825128 + 1.5 Combine like terms: -1.707825128 + 1.5 = -0.207825128 n = -0.207825128 Simplifying n = -0.207825128Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.207825128, -0.207825128}
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