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Simplifying 5400 + 300n + -5n2 = 7000 Solving 5400 + 300n + -5n2 = 7000 Solving for variable 'n'. Reorder the terms: 5400 + -7000 + 300n + -5n2 = 7000 + -7000 Combine like terms: 5400 + -7000 = -1600 -1600 + 300n + -5n2 = 7000 + -7000 Combine like terms: 7000 + -7000 = 0 -1600 + 300n + -5n2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-320 + 60n + -1n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-320 + 60n + -1n2)' equal to zero and attempt to solve: Simplifying -320 + 60n + -1n2 = 0 Solving -320 + 60n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 320 + -60n + n2 = 0 Move the constant term to the right: Add '-320' to each side of the equation. 320 + -60n + -320 + n2 = 0 + -320 Reorder the terms: 320 + -320 + -60n + n2 = 0 + -320 Combine like terms: 320 + -320 = 0 0 + -60n + n2 = 0 + -320 -60n + n2 = 0 + -320 Combine like terms: 0 + -320 = -320 -60n + n2 = -320 The n term is -60n. Take half its coefficient (-30). Square it (900) and add it to both sides. Add '900' to each side of the equation. -60n + 900 + n2 = -320 + 900 Reorder the terms: 900 + -60n + n2 = -320 + 900 Combine like terms: -320 + 900 = 580 900 + -60n + n2 = 580 Factor a perfect square on the left side: (n + -30)(n + -30) = 580 Calculate the square root of the right side: 24.083189158 Break this problem into two subproblems by setting (n + -30) equal to 24.083189158 and -24.083189158.Subproblem 1
n + -30 = 24.083189158 Simplifying n + -30 = 24.083189158 Reorder the terms: -30 + n = 24.083189158 Solving -30 + n = 24.083189158 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '30' to each side of the equation. -30 + 30 + n = 24.083189158 + 30 Combine like terms: -30 + 30 = 0 0 + n = 24.083189158 + 30 n = 24.083189158 + 30 Combine like terms: 24.083189158 + 30 = 54.083189158 n = 54.083189158 Simplifying n = 54.083189158Subproblem 2
n + -30 = -24.083189158 Simplifying n + -30 = -24.083189158 Reorder the terms: -30 + n = -24.083189158 Solving -30 + n = -24.083189158 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '30' to each side of the equation. -30 + 30 + n = -24.083189158 + 30 Combine like terms: -30 + 30 = 0 0 + n = -24.083189158 + 30 n = -24.083189158 + 30 Combine like terms: -24.083189158 + 30 = 5.916810842 n = 5.916810842 Simplifying n = 5.916810842Solution
The solution to the problem is based on the solutions from the subproblems. n = {54.083189158, 5.916810842}Solution
n = {54.083189158, 5.916810842}
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