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Simplifying 14n2 = 12n2 + 40n + 200 Reorder the terms: 14n2 = 200 + 40n + 12n2 Solving 14n2 = 200 + 40n + 12n2 Solving for variable 'n'. Reorder the terms: -200 + -40n + 14n2 + -12n2 = 200 + 40n + 12n2 + -200 + -40n + -12n2 Combine like terms: 14n2 + -12n2 = 2n2 -200 + -40n + 2n2 = 200 + 40n + 12n2 + -200 + -40n + -12n2 Reorder the terms: -200 + -40n + 2n2 = 200 + -200 + 40n + -40n + 12n2 + -12n2 Combine like terms: 200 + -200 = 0 -200 + -40n + 2n2 = 0 + 40n + -40n + 12n2 + -12n2 -200 + -40n + 2n2 = 40n + -40n + 12n2 + -12n2 Combine like terms: 40n + -40n = 0 -200 + -40n + 2n2 = 0 + 12n2 + -12n2 -200 + -40n + 2n2 = 12n2 + -12n2 Combine like terms: 12n2 + -12n2 = 0 -200 + -40n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-100 + -20n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-100 + -20n + n2)' equal to zero and attempt to solve: Simplifying -100 + -20n + n2 = 0 Solving -100 + -20n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '100' to each side of the equation. -100 + -20n + 100 + n2 = 0 + 100 Reorder the terms: -100 + 100 + -20n + n2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + -20n + n2 = 0 + 100 -20n + n2 = 0 + 100 Combine like terms: 0 + 100 = 100 -20n + n2 = 100 The n term is -20n. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20n + 100 + n2 = 100 + 100 Reorder the terms: 100 + -20n + n2 = 100 + 100 Combine like terms: 100 + 100 = 200 100 + -20n + n2 = 200 Factor a perfect square on the left side: (n + -10)(n + -10) = 200 Calculate the square root of the right side: 14.142135624 Break this problem into two subproblems by setting (n + -10) equal to 14.142135624 and -14.142135624.Subproblem 1
n + -10 = 14.142135624 Simplifying n + -10 = 14.142135624 Reorder the terms: -10 + n = 14.142135624 Solving -10 + n = 14.142135624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + n = 14.142135624 + 10 Combine like terms: -10 + 10 = 0 0 + n = 14.142135624 + 10 n = 14.142135624 + 10 Combine like terms: 14.142135624 + 10 = 24.142135624 n = 24.142135624 Simplifying n = 24.142135624Subproblem 2
n + -10 = -14.142135624 Simplifying n + -10 = -14.142135624 Reorder the terms: -10 + n = -14.142135624 Solving -10 + n = -14.142135624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + n = -14.142135624 + 10 Combine like terms: -10 + 10 = 0 0 + n = -14.142135624 + 10 n = -14.142135624 + 10 Combine like terms: -14.142135624 + 10 = -4.142135624 n = -4.142135624 Simplifying n = -4.142135624Solution
The solution to the problem is based on the solutions from the subproblems. n = {24.142135624, -4.142135624}Solution
n = {24.142135624, -4.142135624}
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