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Simplifying 13n2 = 12n2 + -40n + -200 Reorder the terms: 13n2 = -200 + -40n + 12n2 Solving 13n2 = -200 + -40n + 12n2 Solving for variable 'n'. Reorder the terms: 200 + 40n + 13n2 + -12n2 = -200 + -40n + 12n2 + 200 + 40n + -12n2 Combine like terms: 13n2 + -12n2 = 1n2 200 + 40n + 1n2 = -200 + -40n + 12n2 + 200 + 40n + -12n2 Reorder the terms: 200 + 40n + 1n2 = -200 + 200 + -40n + 40n + 12n2 + -12n2 Combine like terms: -200 + 200 = 0 200 + 40n + 1n2 = 0 + -40n + 40n + 12n2 + -12n2 200 + 40n + 1n2 = -40n + 40n + 12n2 + -12n2 Combine like terms: -40n + 40n = 0 200 + 40n + 1n2 = 0 + 12n2 + -12n2 200 + 40n + 1n2 = 12n2 + -12n2 Combine like terms: 12n2 + -12n2 = 0 200 + 40n + 1n2 = 0 Begin completing the square. Move the constant term to the right: Add '-200' to each side of the equation. 200 + 40n + -200 + n2 = 0 + -200 Reorder the terms: 200 + -200 + 40n + n2 = 0 + -200 Combine like terms: 200 + -200 = 0 0 + 40n + n2 = 0 + -200 40n + n2 = 0 + -200 Combine like terms: 0 + -200 = -200 40n + n2 = -200 The n term is 40n. Take half its coefficient (20). Square it (400) and add it to both sides. Add '400' to each side of the equation. 40n + 400 + n2 = -200 + 400 Reorder the terms: 400 + 40n + n2 = -200 + 400 Combine like terms: -200 + 400 = 200 400 + 40n + n2 = 200 Factor a perfect square on the left side: (n + 20)(n + 20) = 200 Calculate the square root of the right side: 14.142135624 Break this problem into two subproblems by setting (n + 20) equal to 14.142135624 and -14.142135624.Subproblem 1
n + 20 = 14.142135624 Simplifying n + 20 = 14.142135624 Reorder the terms: 20 + n = 14.142135624 Solving 20 + n = 14.142135624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-20' to each side of the equation. 20 + -20 + n = 14.142135624 + -20 Combine like terms: 20 + -20 = 0 0 + n = 14.142135624 + -20 n = 14.142135624 + -20 Combine like terms: 14.142135624 + -20 = -5.857864376 n = -5.857864376 Simplifying n = -5.857864376Subproblem 2
n + 20 = -14.142135624 Simplifying n + 20 = -14.142135624 Reorder the terms: 20 + n = -14.142135624 Solving 20 + n = -14.142135624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-20' to each side of the equation. 20 + -20 + n = -14.142135624 + -20 Combine like terms: 20 + -20 = 0 0 + n = -14.142135624 + -20 n = -14.142135624 + -20 Combine like terms: -14.142135624 + -20 = -34.142135624 n = -34.142135624 Simplifying n = -34.142135624Solution
The solution to the problem is based on the solutions from the subproblems. n = {-5.857864376, -34.142135624}
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