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Simplifying 6n + 12 = 4n2 Reorder the terms: 12 + 6n = 4n2 Solving 12 + 6n = 4n2 Solving for variable 'n'. Combine like terms: 4n2 + -4n2 = 0 12 + 6n + -4n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(6 + 3n + -2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + 3n + -2n2)' equal to zero and attempt to solve: Simplifying 6 + 3n + -2n2 = 0 Solving 6 + 3n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3 + -1.5n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -1.5n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + -1.5n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -1.5n + n2 = 0 + 3 -1.5n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 -1.5n + n2 = 3 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 3 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 3 + 0.5625 Combine like terms: 3 + 0.5625 = 3.5625 0.5625 + -1.5n + n2 = 3.5625 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (n + -0.75) equal to 1.887458609 and -1.887458609.Subproblem 1
n + -0.75 = 1.887458609 Simplifying n + -0.75 = 1.887458609 Reorder the terms: -0.75 + n = 1.887458609 Solving -0.75 + n = 1.887458609 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 1.887458609 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.887458609 + 0.75 n = 1.887458609 + 0.75 Combine like terms: 1.887458609 + 0.75 = 2.637458609 n = 2.637458609 Simplifying n = 2.637458609Subproblem 2
n + -0.75 = -1.887458609 Simplifying n + -0.75 = -1.887458609 Reorder the terms: -0.75 + n = -1.887458609 Solving -0.75 + n = -1.887458609 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -1.887458609 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.887458609 + 0.75 n = -1.887458609 + 0.75 Combine like terms: -1.887458609 + 0.75 = -1.137458609 n = -1.137458609 Simplifying n = -1.137458609Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.637458609, -1.137458609}Solution
n = {2.637458609, -1.137458609}
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