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Simplifying 6n2 + 9n = 12 Reorder the terms: 9n + 6n2 = 12 Solving 9n + 6n2 = 12 Solving for variable 'n'. Reorder the terms: -12 + 9n + 6n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 9n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-4 + 3n + 2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-4 + 3n + 2n2)' equal to zero and attempt to solve: Simplifying -4 + 3n + 2n2 = 0 Solving -4 + 3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + 1.5n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.5n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 1.5n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.5n + n2 = 0 + 2 1.5n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.5n + n2 = 2 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 = 2 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 2 + 0.5625 Combine like terms: 2 + 0.5625 = 2.5625 0.5625 + 1.5n + n2 = 2.5625 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (n + 0.75) equal to 1.600781059 and -1.600781059.Subproblem 1
n + 0.75 = 1.600781059 Simplifying n + 0.75 = 1.600781059 Reorder the terms: 0.75 + n = 1.600781059 Solving 0.75 + n = 1.600781059 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 = 1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 1.600781059 + -0.75 n = 1.600781059 + -0.75 Combine like terms: 1.600781059 + -0.75 = 0.850781059 n = 0.850781059 Simplifying n = 0.850781059Subproblem 2
n + 0.75 = -1.600781059 Simplifying n + 0.75 = -1.600781059 Reorder the terms: 0.75 + n = -1.600781059 Solving 0.75 + n = -1.600781059 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 = -1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -1.600781059 + -0.75 n = -1.600781059 + -0.75 Combine like terms: -1.600781059 + -0.75 = -2.350781059 n = -2.350781059 Simplifying n = -2.350781059Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.850781059, -2.350781059}Solution
n = {0.850781059, -2.350781059}
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