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Simplifying n(n + -7) + 3 = 9 Reorder the terms: n(-7 + n) + 3 = 9 (-7 * n + n * n) + 3 = 9 (-7n + n2) + 3 = 9 Reorder the terms: 3 + -7n + n2 = 9 Solving 3 + -7n + n2 = 9 Solving for variable 'n'. Reorder the terms: 3 + -9 + -7n + n2 = 9 + -9 Combine like terms: 3 + -9 = -6 -6 + -7n + n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -6 + -7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + -7n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + -7n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -7n + n2 = 0 + 6 -7n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 -7n + n2 = 6 The n term is -7n. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7n + 12.25 + n2 = 6 + 12.25 Reorder the terms: 12.25 + -7n + n2 = 6 + 12.25 Combine like terms: 6 + 12.25 = 18.25 12.25 + -7n + n2 = 18.25 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (n + -3.5) equal to 4.272001873 and -4.272001873.Subproblem 1
n + -3.5 = 4.272001873 Simplifying n + -3.5 = 4.272001873 Reorder the terms: -3.5 + n = 4.272001873 Solving -3.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = 4.272001873 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 4.272001873 + 3.5 n = 4.272001873 + 3.5 Combine like terms: 4.272001873 + 3.5 = 7.772001873 n = 7.772001873 Simplifying n = 7.772001873Subproblem 2
n + -3.5 = -4.272001873 Simplifying n + -3.5 = -4.272001873 Reorder the terms: -3.5 + n = -4.272001873 Solving -3.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = -4.272001873 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -4.272001873 + 3.5 n = -4.272001873 + 3.5 Combine like terms: -4.272001873 + 3.5 = -0.772001873 n = -0.772001873 Simplifying n = -0.772001873Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.772001873, -0.772001873}
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