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Simplifying n2 + 3n = 162 Reorder the terms: 3n + n2 = 162 Solving 3n + n2 = 162 Solving for variable 'n'. Reorder the terms: -162 + 3n + n2 = 162 + -162 Combine like terms: 162 + -162 = 0 -162 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '162' to each side of the equation. -162 + 3n + 162 + n2 = 0 + 162 Reorder the terms: -162 + 162 + 3n + n2 = 0 + 162 Combine like terms: -162 + 162 = 0 0 + 3n + n2 = 0 + 162 3n + n2 = 0 + 162 Combine like terms: 0 + 162 = 162 3n + n2 = 162 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 162 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 162 + 2.25 Combine like terms: 162 + 2.25 = 164.25 2.25 + 3n + n2 = 164.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 164.25 Calculate the square root of the right side: 12.816005618 Break this problem into two subproblems by setting (n + 1.5) equal to 12.816005618 and -12.816005618.Subproblem 1
n + 1.5 = 12.816005618 Simplifying n + 1.5 = 12.816005618 Reorder the terms: 1.5 + n = 12.816005618 Solving 1.5 + n = 12.816005618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 12.816005618 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 12.816005618 + -1.5 n = 12.816005618 + -1.5 Combine like terms: 12.816005618 + -1.5 = 11.316005618 n = 11.316005618 Simplifying n = 11.316005618Subproblem 2
n + 1.5 = -12.816005618 Simplifying n + 1.5 = -12.816005618 Reorder the terms: 1.5 + n = -12.816005618 Solving 1.5 + n = -12.816005618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -12.816005618 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -12.816005618 + -1.5 n = -12.816005618 + -1.5 Combine like terms: -12.816005618 + -1.5 = -14.316005618 n = -14.316005618 Simplifying n = -14.316005618Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.316005618, -14.316005618}
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