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Simplifying 6n2 = 15 + 12n Solving 6n2 = 15 + 12n Solving for variable 'n'. Reorder the terms: -15 + -12n + 6n2 = 15 + 12n + -15 + -12n Reorder the terms: -15 + -12n + 6n2 = 15 + -15 + 12n + -12n Combine like terms: 15 + -15 = 0 -15 + -12n + 6n2 = 0 + 12n + -12n -15 + -12n + 6n2 = 12n + -12n Combine like terms: 12n + -12n = 0 -15 + -12n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-5 + -4n + 2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-5 + -4n + 2n2)' equal to zero and attempt to solve: Simplifying -5 + -4n + 2n2 = 0 Solving -5 + -4n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + -2n + n2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -2n + 2.5 + n2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -2n + n2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -2n + n2 = 0 + 2.5 -2n + n2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -2n + n2 = 2.5 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 2.5 + 1 Reorder the terms: 1 + -2n + n2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + -2n + n2 = 3.5 Factor a perfect square on the left side: (n + -1)(n + -1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (n + -1) equal to 1.870828693 and -1.870828693.Subproblem 1
n + -1 = 1.870828693 Simplifying n + -1 = 1.870828693 Reorder the terms: -1 + n = 1.870828693 Solving -1 + n = 1.870828693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.870828693 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.870828693 + 1 n = 1.870828693 + 1 Combine like terms: 1.870828693 + 1 = 2.870828693 n = 2.870828693 Simplifying n = 2.870828693Subproblem 2
n + -1 = -1.870828693 Simplifying n + -1 = -1.870828693 Reorder the terms: -1 + n = -1.870828693 Solving -1 + n = -1.870828693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.870828693 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.870828693 + 1 n = -1.870828693 + 1 Combine like terms: -1.870828693 + 1 = -0.870828693 n = -0.870828693 Simplifying n = -0.870828693Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.870828693, -0.870828693}Solution
n = {2.870828693, -0.870828693}
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