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Simplifying 25n2 + -75n + -75n + 225 = 10 Reorder the terms: 225 + -75n + -75n + 25n2 = 10 Combine like terms: -75n + -75n = -150n 225 + -150n + 25n2 = 10 Solving 225 + -150n + 25n2 = 10 Solving for variable 'n'. Reorder the terms: 225 + -10 + -150n + 25n2 = 10 + -10 Combine like terms: 225 + -10 = 215 215 + -150n + 25n2 = 10 + -10 Combine like terms: 10 + -10 = 0 215 + -150n + 25n2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(43 + -30n + 5n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(43 + -30n + 5n2)' equal to zero and attempt to solve: Simplifying 43 + -30n + 5n2 = 0 Solving 43 + -30n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 8.6 + -6n + n2 = 0 Move the constant term to the right: Add '-8.6' to each side of the equation. 8.6 + -6n + -8.6 + n2 = 0 + -8.6 Reorder the terms: 8.6 + -8.6 + -6n + n2 = 0 + -8.6 Combine like terms: 8.6 + -8.6 = 0.0 0.0 + -6n + n2 = 0 + -8.6 -6n + n2 = 0 + -8.6 Combine like terms: 0 + -8.6 = -8.6 -6n + n2 = -8.6 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = -8.6 + 9 Reorder the terms: 9 + -6n + n2 = -8.6 + 9 Combine like terms: -8.6 + 9 = 0.4 9 + -6n + n2 = 0.4 Factor a perfect square on the left side: (n + -3)(n + -3) = 0.4 Calculate the square root of the right side: 0.632455532 Break this problem into two subproblems by setting (n + -3) equal to 0.632455532 and -0.632455532.Subproblem 1
n + -3 = 0.632455532 Simplifying n + -3 = 0.632455532 Reorder the terms: -3 + n = 0.632455532 Solving -3 + n = 0.632455532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 0.632455532 + 3 Combine like terms: -3 + 3 = 0 0 + n = 0.632455532 + 3 n = 0.632455532 + 3 Combine like terms: 0.632455532 + 3 = 3.632455532 n = 3.632455532 Simplifying n = 3.632455532Subproblem 2
n + -3 = -0.632455532 Simplifying n + -3 = -0.632455532 Reorder the terms: -3 + n = -0.632455532 Solving -3 + n = -0.632455532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -0.632455532 + 3 Combine like terms: -3 + 3 = 0 0 + n = -0.632455532 + 3 n = -0.632455532 + 3 Combine like terms: -0.632455532 + 3 = 2.367544468 n = 2.367544468 Simplifying n = 2.367544468Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.632455532, 2.367544468}Solution
n = {3.632455532, 2.367544468}
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