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Simplifying (n + 6)(n + -9) = 90 Reorder the terms: (6 + n)(n + -9) = 90 Reorder the terms: (6 + n)(-9 + n) = 90 Multiply (6 + n) * (-9 + n) (6(-9 + n) + n(-9 + n)) = 90 ((-9 * 6 + n * 6) + n(-9 + n)) = 90 ((-54 + 6n) + n(-9 + n)) = 90 (-54 + 6n + (-9 * n + n * n)) = 90 (-54 + 6n + (-9n + n2)) = 90 Combine like terms: 6n + -9n = -3n (-54 + -3n + n2) = 90 Solving -54 + -3n + n2 = 90 Solving for variable 'n'. Reorder the terms: -54 + -90 + -3n + n2 = 90 + -90 Combine like terms: -54 + -90 = -144 -144 + -3n + n2 = 90 + -90 Combine like terms: 90 + -90 = 0 -144 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '144' to each side of the equation. -144 + -3n + 144 + n2 = 0 + 144 Reorder the terms: -144 + 144 + -3n + n2 = 0 + 144 Combine like terms: -144 + 144 = 0 0 + -3n + n2 = 0 + 144 -3n + n2 = 0 + 144 Combine like terms: 0 + 144 = 144 -3n + n2 = 144 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 = 144 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 144 + 2.25 Combine like terms: 144 + 2.25 = 146.25 2.25 + -3n + n2 = 146.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 146.25 Calculate the square root of the right side: 12.093386622 Break this problem into two subproblems by setting (n + -1.5) equal to 12.093386622 and -12.093386622.Subproblem 1
n + -1.5 = 12.093386622 Simplifying n + -1.5 = 12.093386622 Reorder the terms: -1.5 + n = 12.093386622 Solving -1.5 + n = 12.093386622 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 = 12.093386622 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 12.093386622 + 1.5 n = 12.093386622 + 1.5 Combine like terms: 12.093386622 + 1.5 = 13.593386622 n = 13.593386622 Simplifying n = 13.593386622Subproblem 2
n + -1.5 = -12.093386622 Simplifying n + -1.5 = -12.093386622 Reorder the terms: -1.5 + n = -12.093386622 Solving -1.5 + n = -12.093386622 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 = -12.093386622 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -12.093386622 + 1.5 n = -12.093386622 + 1.5 Combine like terms: -12.093386622 + 1.5 = -10.593386622 n = -10.593386622 Simplifying n = -10.593386622Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.593386622, -10.593386622}
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