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Simplifying (n + 4)(n + -7) = 30 Reorder the terms: (4 + n)(n + -7) = 30 Reorder the terms: (4 + n)(-7 + n) = 30 Multiply (4 + n) * (-7 + n) (4(-7 + n) + n(-7 + n)) = 30 ((-7 * 4 + n * 4) + n(-7 + n)) = 30 ((-28 + 4n) + n(-7 + n)) = 30 (-28 + 4n + (-7 * n + n * n)) = 30 (-28 + 4n + (-7n + n2)) = 30 Combine like terms: 4n + -7n = -3n (-28 + -3n + n2) = 30 Solving -28 + -3n + n2 = 30 Solving for variable 'n'. Reorder the terms: -28 + -30 + -3n + n2 = 30 + -30 Combine like terms: -28 + -30 = -58 -58 + -3n + n2 = 30 + -30 Combine like terms: 30 + -30 = 0 -58 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '58' to each side of the equation. -58 + -3n + 58 + n2 = 0 + 58 Reorder the terms: -58 + 58 + -3n + n2 = 0 + 58 Combine like terms: -58 + 58 = 0 0 + -3n + n2 = 0 + 58 -3n + n2 = 0 + 58 Combine like terms: 0 + 58 = 58 -3n + n2 = 58 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 = 58 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 58 + 2.25 Combine like terms: 58 + 2.25 = 60.25 2.25 + -3n + n2 = 60.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 60.25 Calculate the square root of the right side: 7.762087348 Break this problem into two subproblems by setting (n + -1.5) equal to 7.762087348 and -7.762087348.Subproblem 1
n + -1.5 = 7.762087348 Simplifying n + -1.5 = 7.762087348 Reorder the terms: -1.5 + n = 7.762087348 Solving -1.5 + n = 7.762087348 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 = 7.762087348 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 7.762087348 + 1.5 n = 7.762087348 + 1.5 Combine like terms: 7.762087348 + 1.5 = 9.262087348 n = 9.262087348 Simplifying n = 9.262087348Subproblem 2
n + -1.5 = -7.762087348 Simplifying n + -1.5 = -7.762087348 Reorder the terms: -1.5 + n = -7.762087348 Solving -1.5 + n = -7.762087348 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 = -7.762087348 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -7.762087348 + 1.5 n = -7.762087348 + 1.5 Combine like terms: -7.762087348 + 1.5 = -6.262087348 n = -6.262087348 Simplifying n = -6.262087348Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.262087348, -6.262087348}
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