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Simplifying (n + 4)(n + -5) = 81 Reorder the terms: (4 + n)(n + -5) = 81 Reorder the terms: (4 + n)(-5 + n) = 81 Multiply (4 + n) * (-5 + n) (4(-5 + n) + n(-5 + n)) = 81 ((-5 * 4 + n * 4) + n(-5 + n)) = 81 ((-20 + 4n) + n(-5 + n)) = 81 (-20 + 4n + (-5 * n + n * n)) = 81 (-20 + 4n + (-5n + n2)) = 81 Combine like terms: 4n + -5n = -1n (-20 + -1n + n2) = 81 Solving -20 + -1n + n2 = 81 Solving for variable 'n'. Reorder the terms: -20 + -81 + -1n + n2 = 81 + -81 Combine like terms: -20 + -81 = -101 -101 + -1n + n2 = 81 + -81 Combine like terms: 81 + -81 = 0 -101 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '101' to each side of the equation. -101 + -1n + 101 + n2 = 0 + 101 Reorder the terms: -101 + 101 + -1n + n2 = 0 + 101 Combine like terms: -101 + 101 = 0 0 + -1n + n2 = 0 + 101 -1n + n2 = 0 + 101 Combine like terms: 0 + 101 = 101 -1n + n2 = 101 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 101 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 101 + 0.25 Combine like terms: 101 + 0.25 = 101.25 0.25 + -1n + n2 = 101.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 101.25 Calculate the square root of the right side: 10.062305899 Break this problem into two subproblems by setting (n + -0.5) equal to 10.062305899 and -10.062305899.Subproblem 1
n + -0.5 = 10.062305899 Simplifying n + -0.5 = 10.062305899 Reorder the terms: -0.5 + n = 10.062305899 Solving -0.5 + n = 10.062305899 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 10.062305899 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 10.062305899 + 0.5 n = 10.062305899 + 0.5 Combine like terms: 10.062305899 + 0.5 = 10.562305899 n = 10.562305899 Simplifying n = 10.562305899Subproblem 2
n + -0.5 = -10.062305899 Simplifying n + -0.5 = -10.062305899 Reorder the terms: -0.5 + n = -10.062305899 Solving -0.5 + n = -10.062305899 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -10.062305899 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -10.062305899 + 0.5 n = -10.062305899 + 0.5 Combine like terms: -10.062305899 + 0.5 = -9.562305899 n = -9.562305899 Simplifying n = -9.562305899Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.562305899, -9.562305899}
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