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Simplifying -7 + 3n(n + -9) = 24 Reorder the terms: -7 + 3n(-9 + n) = 24 -7 + (-9 * 3n + n * 3n) = 24 -7 + (-27n + 3n2) = 24 Solving -7 + -27n + 3n2 = 24 Solving for variable 'n'. Reorder the terms: -7 + -24 + -27n + 3n2 = 24 + -24 Combine like terms: -7 + -24 = -31 -31 + -27n + 3n2 = 24 + -24 Combine like terms: 24 + -24 = 0 -31 + -27n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -10.33333333 + -9n + n2 = 0 Move the constant term to the right: Add '10.33333333' to each side of the equation. -10.33333333 + -9n + 10.33333333 + n2 = 0 + 10.33333333 Reorder the terms: -10.33333333 + 10.33333333 + -9n + n2 = 0 + 10.33333333 Combine like terms: -10.33333333 + 10.33333333 = 0.00000000 0.00000000 + -9n + n2 = 0 + 10.33333333 -9n + n2 = 0 + 10.33333333 Combine like terms: 0 + 10.33333333 = 10.33333333 -9n + n2 = 10.33333333 The n term is -9n. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9n + 20.25 + n2 = 10.33333333 + 20.25 Reorder the terms: 20.25 + -9n + n2 = 10.33333333 + 20.25 Combine like terms: 10.33333333 + 20.25 = 30.58333333 20.25 + -9n + n2 = 30.58333333 Factor a perfect square on the left side: (n + -4.5)(n + -4.5) = 30.58333333 Calculate the square root of the right side: 5.530220007 Break this problem into two subproblems by setting (n + -4.5) equal to 5.530220007 and -5.530220007.Subproblem 1
n + -4.5 = 5.530220007 Simplifying n + -4.5 = 5.530220007 Reorder the terms: -4.5 + n = 5.530220007 Solving -4.5 + n = 5.530220007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = 5.530220007 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 5.530220007 + 4.5 n = 5.530220007 + 4.5 Combine like terms: 5.530220007 + 4.5 = 10.030220007 n = 10.030220007 Simplifying n = 10.030220007Subproblem 2
n + -4.5 = -5.530220007 Simplifying n + -4.5 = -5.530220007 Reorder the terms: -4.5 + n = -5.530220007 Solving -4.5 + n = -5.530220007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = -5.530220007 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -5.530220007 + 4.5 n = -5.530220007 + 4.5 Combine like terms: -5.530220007 + 4.5 = -1.030220007 n = -1.030220007 Simplifying n = -1.030220007Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.030220007, -1.030220007}
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