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Simplifying n * n + 6 + -3(n) = 45 Multiply n * n n2 + 6 + -3(n) = 45 Reorder the terms: 6 + -3n + n2 = 45 Solving 6 + -3n + n2 = 45 Solving for variable 'n'. Reorder the terms: 6 + -45 + -3n + n2 = 45 + -45 Combine like terms: 6 + -45 = -39 -39 + -3n + n2 = 45 + -45 Combine like terms: 45 + -45 = 0 -39 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '39' to each side of the equation. -39 + -3n + 39 + n2 = 0 + 39 Reorder the terms: -39 + 39 + -3n + n2 = 0 + 39 Combine like terms: -39 + 39 = 0 0 + -3n + n2 = 0 + 39 -3n + n2 = 0 + 39 Combine like terms: 0 + 39 = 39 -3n + n2 = 39 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 = 39 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 39 + 2.25 Combine like terms: 39 + 2.25 = 41.25 2.25 + -3n + n2 = 41.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 41.25 Calculate the square root of the right side: 6.422616289 Break this problem into two subproblems by setting (n + -1.5) equal to 6.422616289 and -6.422616289.Subproblem 1
n + -1.5 = 6.422616289 Simplifying n + -1.5 = 6.422616289 Reorder the terms: -1.5 + n = 6.422616289 Solving -1.5 + n = 6.422616289 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 = 6.422616289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 6.422616289 + 1.5 n = 6.422616289 + 1.5 Combine like terms: 6.422616289 + 1.5 = 7.922616289 n = 7.922616289 Simplifying n = 7.922616289Subproblem 2
n + -1.5 = -6.422616289 Simplifying n + -1.5 = -6.422616289 Reorder the terms: -1.5 + n = -6.422616289 Solving -1.5 + n = -6.422616289 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 = -6.422616289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -6.422616289 + 1.5 n = -6.422616289 + 1.5 Combine like terms: -6.422616289 + 1.5 = -4.922616289 n = -4.922616289 Simplifying n = -4.922616289Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.922616289, -4.922616289}
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