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Simplifying n(n + 4) + 28 = 28(n + -1)(n + -8) Reorder the terms: n(4 + n) + 28 = 28(n + -1)(n + -8) (4 * n + n * n) + 28 = 28(n + -1)(n + -8) (4n + n2) + 28 = 28(n + -1)(n + -8) Reorder the terms: 28 + 4n + n2 = 28(n + -1)(n + -8) Reorder the terms: 28 + 4n + n2 = 28(-1 + n)(n + -8) Reorder the terms: 28 + 4n + n2 = 28(-1 + n)(-8 + n) Multiply (-1 + n) * (-8 + n) 28 + 4n + n2 = 28(-1(-8 + n) + n(-8 + n)) 28 + 4n + n2 = 28((-8 * -1 + n * -1) + n(-8 + n)) 28 + 4n + n2 = 28((8 + -1n) + n(-8 + n)) 28 + 4n + n2 = 28(8 + -1n + (-8 * n + n * n)) 28 + 4n + n2 = 28(8 + -1n + (-8n + n2)) Combine like terms: -1n + -8n = -9n 28 + 4n + n2 = 28(8 + -9n + n2) 28 + 4n + n2 = (8 * 28 + -9n * 28 + n2 * 28) 28 + 4n + n2 = (224 + -252n + 28n2) Solving 28 + 4n + n2 = 224 + -252n + 28n2 Solving for variable 'n'. Reorder the terms: 28 + -224 + 4n + 252n + n2 + -28n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2 Combine like terms: 28 + -224 = -196 -196 + 4n + 252n + n2 + -28n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2 Combine like terms: 4n + 252n = 256n -196 + 256n + n2 + -28n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2 Combine like terms: n2 + -28n2 = -27n2 -196 + 256n + -27n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2 Reorder the terms: -196 + 256n + -27n2 = 224 + -224 + -252n + 252n + 28n2 + -28n2 Combine like terms: 224 + -224 = 0 -196 + 256n + -27n2 = 0 + -252n + 252n + 28n2 + -28n2 -196 + 256n + -27n2 = -252n + 252n + 28n2 + -28n2 Combine like terms: -252n + 252n = 0 -196 + 256n + -27n2 = 0 + 28n2 + -28n2 -196 + 256n + -27n2 = 28n2 + -28n2 Combine like terms: 28n2 + -28n2 = 0 -196 + 256n + -27n2 = 0 Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. 7.259259259 + -9.481481481n + n2 = 0 Move the constant term to the right: Add '-7.259259259' to each side of the equation. 7.259259259 + -9.481481481n + -7.259259259 + n2 = 0 + -7.259259259 Reorder the terms: 7.259259259 + -7.259259259 + -9.481481481n + n2 = 0 + -7.259259259 Combine like terms: 7.259259259 + -7.259259259 = 0.000000000 0.000000000 + -9.481481481n + n2 = 0 + -7.259259259 -9.481481481n + n2 = 0 + -7.259259259 Combine like terms: 0 + -7.259259259 = -7.259259259 -9.481481481n + n2 = -7.259259259 The n term is -9.481481481n. Take half its coefficient (-4.740740741). Square it (22.47462277) and add it to both sides. Add '22.47462277' to each side of the equation. -9.481481481n + 22.47462277 + n2 = -7.259259259 + 22.47462277 Reorder the terms: 22.47462277 + -9.481481481n + n2 = -7.259259259 + 22.47462277 Combine like terms: -7.259259259 + 22.47462277 = 15.215363511 22.47462277 + -9.481481481n + n2 = 15.215363511 Factor a perfect square on the left side: (n + -4.740740741)(n + -4.740740741) = 15.215363511 Calculate the square root of the right side: 3.900687569 Break this problem into two subproblems by setting (n + -4.740740741) equal to 3.900687569 and -3.900687569.Subproblem 1
n + -4.740740741 = 3.900687569 Simplifying n + -4.740740741 = 3.900687569 Reorder the terms: -4.740740741 + n = 3.900687569 Solving -4.740740741 + n = 3.900687569 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.740740741' to each side of the equation. -4.740740741 + 4.740740741 + n = 3.900687569 + 4.740740741 Combine like terms: -4.740740741 + 4.740740741 = 0.000000000 0.000000000 + n = 3.900687569 + 4.740740741 n = 3.900687569 + 4.740740741 Combine like terms: 3.900687569 + 4.740740741 = 8.64142831 n = 8.64142831 Simplifying n = 8.64142831Subproblem 2
n + -4.740740741 = -3.900687569 Simplifying n + -4.740740741 = -3.900687569 Reorder the terms: -4.740740741 + n = -3.900687569 Solving -4.740740741 + n = -3.900687569 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.740740741' to each side of the equation. -4.740740741 + 4.740740741 + n = -3.900687569 + 4.740740741 Combine like terms: -4.740740741 + 4.740740741 = 0.000000000 0.000000000 + n = -3.900687569 + 4.740740741 n = -3.900687569 + 4.740740741 Combine like terms: -3.900687569 + 4.740740741 = 0.840053172 n = 0.840053172 Simplifying n = 0.840053172Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.64142831, 0.840053172}
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