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Simplifying 3n + 2n(n + -4) = n + 15 Reorder the terms: 3n + 2n(-4 + n) = n + 15 3n + (-4 * 2n + n * 2n) = n + 15 3n + (-8n + 2n2) = n + 15 Combine like terms: 3n + -8n = -5n -5n + 2n2 = n + 15 Reorder the terms: -5n + 2n2 = 15 + n Solving -5n + 2n2 = 15 + n Solving for variable 'n'. Reorder the terms: -15 + -5n + -1n + 2n2 = 15 + n + -15 + -1n Combine like terms: -5n + -1n = -6n -15 + -6n + 2n2 = 15 + n + -15 + -1n Reorder the terms: -15 + -6n + 2n2 = 15 + -15 + n + -1n Combine like terms: 15 + -15 = 0 -15 + -6n + 2n2 = 0 + n + -1n -15 + -6n + 2n2 = n + -1n Combine like terms: n + -1n = 0 -15 + -6n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + -3n + n2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -3n + 7.5 + n2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -3n + n2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -3n + n2 = 0 + 7.5 -3n + n2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -3n + n2 = 7.5 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 = 7.5 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 7.5 + 2.25 Combine like terms: 7.5 + 2.25 = 9.75 2.25 + -3n + n2 = 9.75 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (n + -1.5) equal to 3.122498999 and -3.122498999.Subproblem 1
n + -1.5 = 3.122498999 Simplifying n + -1.5 = 3.122498999 Reorder the terms: -1.5 + n = 3.122498999 Solving -1.5 + n = 3.122498999 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 = 3.122498999 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 3.122498999 + 1.5 n = 3.122498999 + 1.5 Combine like terms: 3.122498999 + 1.5 = 4.622498999 n = 4.622498999 Simplifying n = 4.622498999Subproblem 2
n + -1.5 = -3.122498999 Simplifying n + -1.5 = -3.122498999 Reorder the terms: -1.5 + n = -3.122498999 Solving -1.5 + n = -3.122498999 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 = -3.122498999 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -3.122498999 + 1.5 n = -3.122498999 + 1.5 Combine like terms: -3.122498999 + 1.5 = -1.622498999 n = -1.622498999 Simplifying n = -1.622498999Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.622498999, -1.622498999}
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