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Simplifying 14n2 + -144n + 112 = 0 Reorder the terms: 112 + -144n + 14n2 = 0 Solving 112 + -144n + 14n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(56 + -72n + 7n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(56 + -72n + 7n2)' equal to zero and attempt to solve: Simplifying 56 + -72n + 7n2 = 0 Solving 56 + -72n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 8 + -10.28571429n + n2 = 0 Move the constant term to the right: Add '-8' to each side of the equation. 8 + -10.28571429n + -8 + n2 = 0 + -8 Reorder the terms: 8 + -8 + -10.28571429n + n2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -10.28571429n + n2 = 0 + -8 -10.28571429n + n2 = 0 + -8 Combine like terms: 0 + -8 = -8 -10.28571429n + n2 = -8 The n term is -10.28571429n. Take half its coefficient (-5.142857145). Square it (26.44897961) and add it to both sides. Add '26.44897961' to each side of the equation. -10.28571429n + 26.44897961 + n2 = -8 + 26.44897961 Reorder the terms: 26.44897961 + -10.28571429n + n2 = -8 + 26.44897961 Combine like terms: -8 + 26.44897961 = 18.44897961 26.44897961 + -10.28571429n + n2 = 18.44897961 Factor a perfect square on the left side: (n + -5.142857145)(n + -5.142857145) = 18.44897961 Calculate the square root of the right side: 4.295227539 Break this problem into two subproblems by setting (n + -5.142857145) equal to 4.295227539 and -4.295227539.Subproblem 1
n + -5.142857145 = 4.295227539 Simplifying n + -5.142857145 = 4.295227539 Reorder the terms: -5.142857145 + n = 4.295227539 Solving -5.142857145 + n = 4.295227539 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.142857145' to each side of the equation. -5.142857145 + 5.142857145 + n = 4.295227539 + 5.142857145 Combine like terms: -5.142857145 + 5.142857145 = 0.000000000 0.000000000 + n = 4.295227539 + 5.142857145 n = 4.295227539 + 5.142857145 Combine like terms: 4.295227539 + 5.142857145 = 9.438084684 n = 9.438084684 Simplifying n = 9.438084684Subproblem 2
n + -5.142857145 = -4.295227539 Simplifying n + -5.142857145 = -4.295227539 Reorder the terms: -5.142857145 + n = -4.295227539 Solving -5.142857145 + n = -4.295227539 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.142857145' to each side of the equation. -5.142857145 + 5.142857145 + n = -4.295227539 + 5.142857145 Combine like terms: -5.142857145 + 5.142857145 = 0.000000000 0.000000000 + n = -4.295227539 + 5.142857145 n = -4.295227539 + 5.142857145 Combine like terms: -4.295227539 + 5.142857145 = 0.847629606 n = 0.847629606 Simplifying n = 0.847629606Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.438084684, 0.847629606}Solution
n = {9.438084684, 0.847629606}
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