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Simplifying -24n2 + 60n + -30 = 0 Reorder the terms: -30 + 60n + -24n2 = 0 Solving -30 + 60n + -24n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '6'. 6(-5 + 10n + -4n2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-5 + 10n + -4n2)' equal to zero and attempt to solve: Simplifying -5 + 10n + -4n2 = 0 Solving -5 + 10n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 1.25 + -2.5n + n2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + -2.5n + -1.25 + n2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + -2.5n + n2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + -2.5n + n2 = 0 + -1.25 -2.5n + n2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 -2.5n + n2 = -1.25 The n term is -2.5n. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5n + 1.5625 + n2 = -1.25 + 1.5625 Reorder the terms: 1.5625 + -2.5n + n2 = -1.25 + 1.5625 Combine like terms: -1.25 + 1.5625 = 0.3125 1.5625 + -2.5n + n2 = 0.3125 Factor a perfect square on the left side: (n + -1.25)(n + -1.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (n + -1.25) equal to 0.559016994 and -0.559016994.Subproblem 1
n + -1.25 = 0.559016994 Simplifying n + -1.25 = 0.559016994 Reorder the terms: -1.25 + n = 0.559016994 Solving -1.25 + n = 0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + n = 0.559016994 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = 0.559016994 + 1.25 n = 0.559016994 + 1.25 Combine like terms: 0.559016994 + 1.25 = 1.809016994 n = 1.809016994 Simplifying n = 1.809016994Subproblem 2
n + -1.25 = -0.559016994 Simplifying n + -1.25 = -0.559016994 Reorder the terms: -1.25 + n = -0.559016994 Solving -1.25 + n = -0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + n = -0.559016994 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = -0.559016994 + 1.25 n = -0.559016994 + 1.25 Combine like terms: -0.559016994 + 1.25 = 0.690983006 n = 0.690983006 Simplifying n = 0.690983006Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.809016994, 0.690983006}Solution
n = {1.809016994, 0.690983006}
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