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Simplifying 6n2 + 72n = 192 Reorder the terms: 72n + 6n2 = 192 Solving 72n + 6n2 = 192 Solving for variable 'n'. Reorder the terms: -192 + 72n + 6n2 = 192 + -192 Combine like terms: 192 + -192 = 0 -192 + 72n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-32 + 12n + n2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-32 + 12n + n2)' equal to zero and attempt to solve: Simplifying -32 + 12n + n2 = 0 Solving -32 + 12n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + 12n + 32 + n2 = 0 + 32 Reorder the terms: -32 + 32 + 12n + n2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 12n + n2 = 0 + 32 12n + n2 = 0 + 32 Combine like terms: 0 + 32 = 32 12n + n2 = 32 The n term is 12n. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12n + 36 + n2 = 32 + 36 Reorder the terms: 36 + 12n + n2 = 32 + 36 Combine like terms: 32 + 36 = 68 36 + 12n + n2 = 68 Factor a perfect square on the left side: (n + 6)(n + 6) = 68 Calculate the square root of the right side: 8.246211251 Break this problem into two subproblems by setting (n + 6) equal to 8.246211251 and -8.246211251.Subproblem 1
n + 6 = 8.246211251 Simplifying n + 6 = 8.246211251 Reorder the terms: 6 + n = 8.246211251 Solving 6 + n = 8.246211251 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = 8.246211251 + -6 Combine like terms: 6 + -6 = 0 0 + n = 8.246211251 + -6 n = 8.246211251 + -6 Combine like terms: 8.246211251 + -6 = 2.246211251 n = 2.246211251 Simplifying n = 2.246211251Subproblem 2
n + 6 = -8.246211251 Simplifying n + 6 = -8.246211251 Reorder the terms: 6 + n = -8.246211251 Solving 6 + n = -8.246211251 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = -8.246211251 + -6 Combine like terms: 6 + -6 = 0 0 + n = -8.246211251 + -6 n = -8.246211251 + -6 Combine like terms: -8.246211251 + -6 = -14.246211251 n = -14.246211251 Simplifying n = -14.246211251Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.246211251, -14.246211251}Solution
n = {2.246211251, -14.246211251}
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