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Simplifying 800 = 16n2 + 56n Reorder the terms: 800 = 56n + 16n2 Solving 800 = 56n + 16n2 Solving for variable 'n'. Reorder the terms: 800 + -56n + -16n2 = 56n + -56n + 16n2 + -16n2 Combine like terms: 56n + -56n = 0 800 + -56n + -16n2 = 0 + 16n2 + -16n2 800 + -56n + -16n2 = 16n2 + -16n2 Combine like terms: 16n2 + -16n2 = 0 800 + -56n + -16n2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(100 + -7n + -2n2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(100 + -7n + -2n2)' equal to zero and attempt to solve: Simplifying 100 + -7n + -2n2 = 0 Solving 100 + -7n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -50 + 3.5n + n2 = 0 Move the constant term to the right: Add '50' to each side of the equation. -50 + 3.5n + 50 + n2 = 0 + 50 Reorder the terms: -50 + 50 + 3.5n + n2 = 0 + 50 Combine like terms: -50 + 50 = 0 0 + 3.5n + n2 = 0 + 50 3.5n + n2 = 0 + 50 Combine like terms: 0 + 50 = 50 3.5n + n2 = 50 The n term is 3.5n. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5n + 3.0625 + n2 = 50 + 3.0625 Reorder the terms: 3.0625 + 3.5n + n2 = 50 + 3.0625 Combine like terms: 50 + 3.0625 = 53.0625 3.0625 + 3.5n + n2 = 53.0625 Factor a perfect square on the left side: (n + 1.75)(n + 1.75) = 53.0625 Calculate the square root of the right side: 7.284401142 Break this problem into two subproblems by setting (n + 1.75) equal to 7.284401142 and -7.284401142.Subproblem 1
n + 1.75 = 7.284401142 Simplifying n + 1.75 = 7.284401142 Reorder the terms: 1.75 + n = 7.284401142 Solving 1.75 + n = 7.284401142 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 7.284401142 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 7.284401142 + -1.75 n = 7.284401142 + -1.75 Combine like terms: 7.284401142 + -1.75 = 5.534401142 n = 5.534401142 Simplifying n = 5.534401142Subproblem 2
n + 1.75 = -7.284401142 Simplifying n + 1.75 = -7.284401142 Reorder the terms: 1.75 + n = -7.284401142 Solving 1.75 + n = -7.284401142 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -7.284401142 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -7.284401142 + -1.75 n = -7.284401142 + -1.75 Combine like terms: -7.284401142 + -1.75 = -9.034401142 n = -9.034401142 Simplifying n = -9.034401142Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.534401142, -9.034401142}Solution
n = {5.534401142, -9.034401142}
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