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Simplifying 4n2 = -12n + -3 Reorder the terms: 4n2 = -3 + -12n Solving 4n2 = -3 + -12n Solving for variable 'n'. Reorder the terms: 3 + 12n + 4n2 = -3 + -12n + 3 + 12n Reorder the terms: 3 + 12n + 4n2 = -3 + 3 + -12n + 12n Combine like terms: -3 + 3 = 0 3 + 12n + 4n2 = 0 + -12n + 12n 3 + 12n + 4n2 = -12n + 12n Combine like terms: -12n + 12n = 0 3 + 12n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 3n + n2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 3n + -0.75 + n2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 3n + n2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 3n + n2 = 0 + -0.75 3n + n2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 3n + n2 = -0.75 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 = -0.75 + 2.25 Reorder the terms: 2.25 + 3n + n2 = -0.75 + 2.25 Combine like terms: -0.75 + 2.25 = 1.5 2.25 + 3n + n2 = 1.5 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (n + 1.5) equal to 1.224744871 and -1.224744871.Subproblem 1
n + 1.5 = 1.224744871 Simplifying n + 1.5 = 1.224744871 Reorder the terms: 1.5 + n = 1.224744871 Solving 1.5 + n = 1.224744871 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 = 1.224744871 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 1.224744871 + -1.5 n = 1.224744871 + -1.5 Combine like terms: 1.224744871 + -1.5 = -0.275255129 n = -0.275255129 Simplifying n = -0.275255129Subproblem 2
n + 1.5 = -1.224744871 Simplifying n + 1.5 = -1.224744871 Reorder the terms: 1.5 + n = -1.224744871 Solving 1.5 + n = -1.224744871 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 = -1.224744871 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -1.224744871 + -1.5 n = -1.224744871 + -1.5 Combine like terms: -1.224744871 + -1.5 = -2.724744871 n = -2.724744871 Simplifying n = -2.724744871Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.275255129, -2.724744871}
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