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Simplifying 12n2 = 15 + 12n Solving 12n2 = 15 + 12n Solving for variable 'n'. Reorder the terms: -15 + -12n + 12n2 = 15 + 12n + -15 + -12n Reorder the terms: -15 + -12n + 12n2 = 15 + -15 + 12n + -12n Combine like terms: 15 + -15 = 0 -15 + -12n + 12n2 = 0 + 12n + -12n -15 + -12n + 12n2 = 12n + -12n Combine like terms: 12n + -12n = 0 -15 + -12n + 12n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-5 + -4n + 4n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-5 + -4n + 4n2)' equal to zero and attempt to solve: Simplifying -5 + -4n + 4n2 = 0 Solving -5 + -4n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + -1n + n2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -1n + 1.25 + n2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -1n + n2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -1n + n2 = 0 + 1.25 -1n + n2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -1n + n2 = 1.25 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 1.25 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 1.25 + 0.25 Combine like terms: 1.25 + 0.25 = 1.5 0.25 + -1n + n2 = 1.5 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (n + -0.5) equal to 1.224744871 and -1.224744871.Subproblem 1
n + -0.5 = 1.224744871 Simplifying n + -0.5 = 1.224744871 Reorder the terms: -0.5 + n = 1.224744871 Solving -0.5 + n = 1.224744871 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.224744871 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.224744871 + 0.5 n = 1.224744871 + 0.5 Combine like terms: 1.224744871 + 0.5 = 1.724744871 n = 1.724744871 Simplifying n = 1.724744871Subproblem 2
n + -0.5 = -1.224744871 Simplifying n + -0.5 = -1.224744871 Reorder the terms: -0.5 + n = -1.224744871 Solving -0.5 + n = -1.224744871 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.224744871 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.224744871 + 0.5 n = -1.224744871 + 0.5 Combine like terms: -1.224744871 + 0.5 = -0.724744871 n = -0.724744871 Simplifying n = -0.724744871Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.724744871, -0.724744871}Solution
n = {1.724744871, -0.724744871}
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