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Simplifying n + (3n + -15)(n + 20) = 0 Reorder the terms: n + (-15 + 3n)(n + 20) = 0 Reorder the terms: n + (-15 + 3n)(20 + n) = 0 Multiply (-15 + 3n) * (20 + n) n + (-15(20 + n) + 3n * (20 + n)) = 0 n + ((20 * -15 + n * -15) + 3n * (20 + n)) = 0 n + ((-300 + -15n) + 3n * (20 + n)) = 0 n + (-300 + -15n + (20 * 3n + n * 3n)) = 0 n + (-300 + -15n + (60n + 3n2)) = 0 Combine like terms: -15n + 60n = 45n n + (-300 + 45n + 3n2) = 0 Reorder the terms: -300 + n + 45n + 3n2 = 0 Combine like terms: n + 45n = 46n -300 + 46n + 3n2 = 0 Solving -300 + 46n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -100 + 15.33333333n + n2 = 0 Move the constant term to the right: Add '100' to each side of the equation. -100 + 15.33333333n + 100 + n2 = 0 + 100 Reorder the terms: -100 + 100 + 15.33333333n + n2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + 15.33333333n + n2 = 0 + 100 15.33333333n + n2 = 0 + 100 Combine like terms: 0 + 100 = 100 15.33333333n + n2 = 100 The n term is 15.33333333n. Take half its coefficient (7.666666665). Square it (58.77777775) and add it to both sides. Add '58.77777775' to each side of the equation. 15.33333333n + 58.77777775 + n2 = 100 + 58.77777775 Reorder the terms: 58.77777775 + 15.33333333n + n2 = 100 + 58.77777775 Combine like terms: 100 + 58.77777775 = 158.77777775 58.77777775 + 15.33333333n + n2 = 158.77777775 Factor a perfect square on the left side: (n + 7.666666665)(n + 7.666666665) = 158.77777775 Calculate the square root of the right side: 12.600705447 Break this problem into two subproblems by setting (n + 7.666666665) equal to 12.600705447 and -12.600705447.Subproblem 1
n + 7.666666665 = 12.600705447 Simplifying n + 7.666666665 = 12.600705447 Reorder the terms: 7.666666665 + n = 12.600705447 Solving 7.666666665 + n = 12.600705447 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.666666665' to each side of the equation. 7.666666665 + -7.666666665 + n = 12.600705447 + -7.666666665 Combine like terms: 7.666666665 + -7.666666665 = 0.000000000 0.000000000 + n = 12.600705447 + -7.666666665 n = 12.600705447 + -7.666666665 Combine like terms: 12.600705447 + -7.666666665 = 4.934038782 n = 4.934038782 Simplifying n = 4.934038782Subproblem 2
n + 7.666666665 = -12.600705447 Simplifying n + 7.666666665 = -12.600705447 Reorder the terms: 7.666666665 + n = -12.600705447 Solving 7.666666665 + n = -12.600705447 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.666666665' to each side of the equation. 7.666666665 + -7.666666665 + n = -12.600705447 + -7.666666665 Combine like terms: 7.666666665 + -7.666666665 = 0.000000000 0.000000000 + n = -12.600705447 + -7.666666665 n = -12.600705447 + -7.666666665 Combine like terms: -12.600705447 + -7.666666665 = -20.267372112 n = -20.267372112 Simplifying n = -20.267372112Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.934038782, -20.267372112}
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