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Simplifying -4n2 + 68n = 72 Reorder the terms: 68n + -4n2 = 72 Solving 68n + -4n2 = 72 Solving for variable 'n'. Reorder the terms: -72 + 68n + -4n2 = 72 + -72 Combine like terms: 72 + -72 = 0 -72 + 68n + -4n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-18 + 17n + -1n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-18 + 17n + -1n2)' equal to zero and attempt to solve: Simplifying -18 + 17n + -1n2 = 0 Solving -18 + 17n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 18 + -17n + n2 = 0 Move the constant term to the right: Add '-18' to each side of the equation. 18 + -17n + -18 + n2 = 0 + -18 Reorder the terms: 18 + -18 + -17n + n2 = 0 + -18 Combine like terms: 18 + -18 = 0 0 + -17n + n2 = 0 + -18 -17n + n2 = 0 + -18 Combine like terms: 0 + -18 = -18 -17n + n2 = -18 The n term is -17n. Take half its coefficient (-8.5). Square it (72.25) and add it to both sides. Add '72.25' to each side of the equation. -17n + 72.25 + n2 = -18 + 72.25 Reorder the terms: 72.25 + -17n + n2 = -18 + 72.25 Combine like terms: -18 + 72.25 = 54.25 72.25 + -17n + n2 = 54.25 Factor a perfect square on the left side: (n + -8.5)(n + -8.5) = 54.25 Calculate the square root of the right side: 7.365459931 Break this problem into two subproblems by setting (n + -8.5) equal to 7.365459931 and -7.365459931.Subproblem 1
n + -8.5 = 7.365459931 Simplifying n + -8.5 = 7.365459931 Reorder the terms: -8.5 + n = 7.365459931 Solving -8.5 + n = 7.365459931 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + n = 7.365459931 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + n = 7.365459931 + 8.5 n = 7.365459931 + 8.5 Combine like terms: 7.365459931 + 8.5 = 15.865459931 n = 15.865459931 Simplifying n = 15.865459931Subproblem 2
n + -8.5 = -7.365459931 Simplifying n + -8.5 = -7.365459931 Reorder the terms: -8.5 + n = -7.365459931 Solving -8.5 + n = -7.365459931 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + n = -7.365459931 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + n = -7.365459931 + 8.5 n = -7.365459931 + 8.5 Combine like terms: -7.365459931 + 8.5 = 1.134540069 n = 1.134540069 Simplifying n = 1.134540069Solution
The solution to the problem is based on the solutions from the subproblems. n = {15.865459931, 1.134540069}Solution
n = {15.865459931, 1.134540069}
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